- Email: [email protected]
Simulation of atomic mobilities, interdiffusivities and diffusional evolution in fcc Ni-Cu-Ti alloys
Accepted Manuscript Simulation of atomic mobilities, interdiffusivities and diffusional evolution in fcc Ni-CuTi alloys Chunlin Li, Shenming Huang, Yajun Liu PII:
S0925-8388(18)33926-4
DOI:
https://doi.org/10.1016/j.jallcom.2018.10.251
Reference:
JALCOM 48069
To appear in:
Journal of Alloys and Compounds
Received Date: 4 July 2018 Revised Date:
17 October 2018
Accepted Date: 21 October 2018
Please cite this article as: C. Li, S. Huang, Y. Liu, Simulation of atomic mobilities, interdiffusivities and diffusional evolution in fcc Ni-Cu-Ti alloys, Journal of Alloys and Compounds (2018), doi: https:// doi.org/10.1016/j.jallcom.2018.10.251. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Simulation of Atomic Mobilities, Interdiffusivities
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Authors:
SC
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and Diffusional Evolution in fcc Ni-Cu-Ti Alloys
Chunlin Lia*, Shenming Huangb, Yajun Liub a
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Nanjing Institute of Electronic Technology, Nanjing, Jiangshu, 210039, P. R. China b* School of Materials and Energy, Guangdong University of Technology, Guangzhou, Guangdong, 510006, P. R. China
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* Corresponding Author Tel: 86-10-13851990477
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Email: [email protected], and [email protected]
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Simulation of Atomic Mobilities, Interdiffusivities and Diffusional Evolution in fcc Ni-Cu-Ti Alloys
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Abstract
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As a contribution to establish a general Ni-based mobility database, the atomic mobilities in fcc Ni-Cu-Ti alloys were explored in this work. Diffusion couples were annealed at 1273 K for 150 hrs, and the interdiffusion coefficients were evaluated at common composition points of diffusion profiles. By combining such interdiffusion coefficients and the available
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thermodynamic information, the atomic mobilities of Ni, Cu and Ti in fcc Ni-Cu-Ti alloys were assessed as functions of temperature and compositions within the CALPHAD framework. Further verification of the obtained atomic mobilities was conducted by comparing the calculated and measured interdiffusvities, diffusion profiles in diffusion couples and diffusion paths in the Gibbs triangle. Keywords: Mobility; Diffusion; fcc Ni-Cu-Ti alloys; CALPHAD
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1. Introduction
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Characterized by high specific strength, excellent resistance against oxidation and corrosion, Ni-based superalloys are widely used in gas turbine blades and vanes for power plants in high-temperature applications [1-2]. Principally, alloy microstructures can be significantly affected by diffusion process for multicomponent systems, and the knowledge from materials kinetics can be used to optimize alloy performance at high temperatures [3]. In the past decades, studies on alloy kinetics are driven by increasing demands on
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computational simulation of microstructual evolutions, where the phase field approach is one representative technique. Based upon the CALPHAD method, DICTRA was developed to provide fundamental solutions for diffusion problems [4]. DICTRA is capable of simulating diffusion-controlled problems, such as dissolution, solidification, carburization, coarsening and growth. Up to now, commercial thermodynamic databases for titanium alloys are available, and the atomic mobility descriptions for some binary and ternary subsystems have been published. The Ni-Cu-Ti ternary system plays an essential role in exploring Ni-based superalloys. However, related kinetic description based on the CALPHAD framework is still missing. Thus, the objective of this work is to explore accurate mobility description for Ni-Cu-Ti alloys so that a multicomponent Ni-based kinetic database can be established, which indeed provides further insights into diffusion characteristics. 1
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2. Diffusion Methodology 2.1 Model Description
Mi =
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Based on the absolute reaction rate theory, the atomic mobility parameter of element i can be divided into a frequency term and an activation enthalpy term, which can be expressed as [5]: −Q* + RT ln( M i0 ) Φ 1 1 exp( i )= exp( i ) RT RT RT RT
(1)
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, where M i stands for atomic mobility for element i ; M i0 is the frequency factor; Qi*
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denotes the activation enthalpy; R is the gas constant; T is the absolute temperature. Φ i is equivalent to −Qi* + RT ln( M i0 ) , which can be expanded by the Redlich-Kister polynomial as [6]: Φ i = ∑ x j Φ ij + ∑∑ x j xk [∑ r Φ ij ,k ( x j − xk ) r ] + ∑∑∑ x j xk xl [ν sjkl s Φ ij , k ,l ] j
j
k> j
r
j
(2)
k > j l >k
, where x j is the mole fraction of element i ; Φ ij stands for the mobility end-member of element i to diffuse in j ;
r
Φ ij ,k denotes the interaction parameters for element i to
diffuse in j - k binary systems;
s
Φ ij , k ,l is the ternary interaction parameter for element i ;
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the parameter ν sjkl can be expressed as [6]:
ν sjkl = xs + (1 − xi − x j − xk ) / 3
(3)
According to the Einstein relation, the tracer diffusion coefficients of element i , Di* , can
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be related to its atomic mobility by [7]:
Di* = RTM i
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(4) rN In a number-fixed frame of reference, the interdiffusion flux of element i , J i , can be expressed as: n −1 r J iN = −∑ D% ijn ∇C j
(5)
j =1
, where the superscript N is the number-fixed frame of reference; C j denotes for the ~ volume concentration of element j ; Dijn is the interdiffusion coefficient, which relates the flux of element i to the concentration gradient of element j ; element n is chosen as the dependent element. The interdiffusion coefficient D% n can be expressed as follows [8]: ij
∂µ ∂µ D% ijn = ∑ [(δ ik − xk ) xi M i ( i − i )] ∂x j ∂xn i
(6)
, where the superscript n is the dependent element in the diffusion system, i and j stand 2
ACCEPTED MANUSCRIPT for Cu or Ti, δ ik denotes the Kronecker delta ( δ ik = 0 if i≠k; or δ ik = 1 ); µi is the chemical potential of element i . Provided that the interdiffusion fluxes of all the elements are known, the spatial and temporal evolution of element i can be described by the equation of continuity as follows: ~ ∂Ci + ∇ ⋅ J iN = 0 ∂t
(7)
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, where t is time. DICTRA is a software package based upon the finite difference algorithm, which can be used to solve Eqn. 7 with suitable initial and boundary conditions.
2.2 Evaluation of Interdiffusion Coefficients of element i can be directly evaluated as follows [9]:
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From the concentration profiles developed in diffusion couples, the interdiffusion flux
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r 1 Ci ( x ) J iN = ∫ − + ( x − x0 )dCi 2t Ci or Ci
(8)
, where x0 is the location for the Matano plane. The combination of Eqns. 5 and 8 leads to the following constrains that can be used to extract interdiffusion coefficients at common compositions of diffusion couples:
∫
CCu ( x )
∫
CTi ( x )
CTi−
Ni ∂CCu % Ni ∂CTi ( x − x0 )dCCu = −2t D% CuCu + DCuTi ∂x ∂x
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− + CCu or CCu
or
CTi+
Ni ∂CCu % Ni ∂CTi ( x − x0 )dCTi = −2t D% TiCu + DTiTi ∂x ∂x
(9)
(10)
, where Ni is chosen as the dependent element.
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According to the thermodynamic stability constraints from solid solutions, all the obtained interdiffusion coefficients should be further validated by the following equations
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[10]:
Ni Ni D% CuCu + D% TiTi >0 Ni Ni Ni % Ni D% CuCu D% TiTi − D% CuTi DTiCu ≥ 0 Ni Ni 2 Ni % Ni ( D% CuCu − D% TiTi ) + 4 D% CuTi DTiCu ≥ 0
(11) (12) (13)
3. Experimental Copper (purity: 99.6 wt%), Titanium (purity: 99.9 wt.%), and Nickel (purity: 99.5 wt.%) were used as starting materials to prepared needed alloys. The compositions of such alloys are designed so that the diffusion paths superimposed on the Gibbs triangle can cover a great majority of the fcc phase region. The compositions of designed alloys are listed in Table1, 3
ACCEPTED MANUSCRIPT and the combination manners to form diffusion couples are also given in Table 1. All the alloys were melted in alumina crucibles in an induction melter with a vacuum condition, and the ingots were then homogenized at 1273 K for 150 hrs. After that, the ingots were cut by diamond saw to obtain suitable sized blocks (~10×10×3 mm3), which were then grounded with SiC carbide papers down to 0.5 µm and further polished with diamond slurry (diamond
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size: ~0.25 µm). Seven diffusion couples were prepared by binding the polished surfaces of two blocks together, which were then annealed at 1273 K for 150 hrs in a hot-pressing furnace with an external load of 5 MPa. All the diffusion annealing was carried out under a vacuum atmosphere. In a subsequent step, the annealed diffusion couples were cut parallel to
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the diffusion direction, followed by standard metallographic preparation. The last step is to obtain concentration profiles by means of electron probe microanalysis technique (EPMA) on the polished section.
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4. Results and Discussion
In order to calculate the interdiffusivities by Eqn. 9 and 10, the location of intersection of diffusion couples must be known. Such points were obtained as follows: Firstly, the
TiTi
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experimental concentration curves for diffusion couples were fitted by the highly non-linear equations given in the work of Bouchet et al. [11]. Secondly, such fitted concentration curves were plotted in a Gibbs triangle to locate all the intersection points of diffusion Ni couples. Based upon Eqns. 9 and 10, the two main interdiffusion coefficients, D% CuCu and Ni Ni Ni % % % D , and the two cross interdiffusion coefficients, D and D , can be evaluated at CuTi
TiCu
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the intersections of diffusion paths. Such experimentally-measured interdiffusivities were then used to retrieve the atomic mobilities of Cu, Ti and Ni in fcc Ni-Cu-Ti alloys in the Parrot module in DICTRA. Thermodynamic parameters of fcc Ni-Cu-Ti alloys optimized
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by Zhu et al. [12] are used in the present work, and the calculated phase diagram for 1273 K is presented in Fig. 1. It is noted that the experimentally obtained interdiffusion coefficients are dependent upon thermodynamic information, and the atomic mobilities obtained are independent of thermodynamic factors. Accordingly, such atomic mobilities can be stored in purely kinetic databases in the CALPHAD framework. The obtained interaction parameters and mobility end-members are presented in Table 2, along with those published in the literature [13-14]. Compiled in Table 3 are the calculated and experimentally-measured interdiffusion coefficients at 1273 K, where the agreement is satisfactory. In addition, all the experimentally-measured interdiffusion coefficients obey the stability condition given in Ni Ni Ni Ni Eqns. 11–13. It is obvious that D% CuCu , D% TiTi , D% CuTi and D% TiCu in Table 3 are all Ni Ni Ni % , D% CuTi characterized with positive sign. In addition, DTiTi is generally larger than D% CuCu 4
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The quality of the obtained atomic mobilities can be further validated by simulation diffusion evolution in diffusion couples, when the atomic mobilities are combined with related thermodynamic description. Figs. 2-8 show the experimentally-measured and DICTRA-predicted concentration profiles for Ni-Cu-Ti diffusion couples annealed at 1273 K
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for 150 hrs, in which the Matano plane is superimposed as vertical lines. In general, the calculated results agree well with the corresponding experimentally measured ones, and the concentration curves in such plots generally show a monotonous nature. A comparison between the calculated and experimentally-measured diffusion paths for diffusion couples
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annealed at 1273 K for 150 hrs is presented in Figs. 9. As shown in the figure, all of the diffusion paths developed in the diffusion couples show S-shaped geometry, and there is a good agreement between experimental and predicted calculated diffusion paths.
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5. Conclusions
Based upon the experimentally-measured interdiffusivities and the thermodynamic description for fcc Ni-Cu-Ti alloys, the atomic mobilities of Ni, Cu and Ti for fcc Ni-Cu-Ti alloys were critically assessed in DICTRA. Satisfactory agreement was obtained from comprehensive comparisons made between the calculated and experimental interdiffusivities.
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diffusion couples.
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In order to further validate the quality of the present atomic mobilities, comparison between experimental and model-predicted concentration profiles in diffusion couples was made, and the agreement is satisfactory. In addition, diffusion paths are presented in the Gibbs triangle, which are of great help to gain insights into investigated spatial and temporal evolution in
5
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References: [1] A. Tamm, A.Aabloo, M. Klintenberg, M. Stocks, A. Caro, Acta Mater., 99, 2015,
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307-312. [2] A. C. Coppa, M. Kappor, R. Noebe, G. B. Thompson, Intermetallics, 67, 2015, 56-62. [3] Y. Nishida, Y. Tsukada, T. Koyama, M. Kurata, J. Nuclear Mater., 466, 2015, 551-559. [4] A. Borgenstam, L. Hoglund, J. Agren, A. Engstrom, J. Phase Equilib. Diff., 21, 2000,
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269-280. [5] J. O. Andersson, J. Ågren, J. Appl. Phys., 72, 1992, 1350-1355. [6] X. He, W. Zhang, M. Yan, C. Chen, Y. Du, L. Zhang, B. Huang, CALPHAD, 49, 2015, 35-40.
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[7] Y. Liu, T. Pan, L. Zhang, D. Yu, Y. Ge, J. Alloys Compd., 476, 2009, 429-435. [8] Z. Zhou, Y. Liu, G. Sheng, F. Lei, Z. Kang, CALPHAD, 48, 2015, 151-156. [9] M. A. Dayananda, J. Phase Equi. Diff., 26, 2005, 441-446. [10] J. S. Kirkaldy, G. R. Purdy, Can. J. Phys., 47, 1969, 865-871. [11] R. Bouchet, R. Mevrel, Acta Mater., 50(19), 2002, 4887-4900. [12] W. J. Zhu, L. I. Duarte. C. Leinenbach, CALPHAD, 47, 2014, 9-22. [13] J. Chen, Y. Liu, F. Lei, G. Sheng, Z. Kang, CALPHAD, 47, 2014, 123-128.
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[14] M. Liu, L. Zhang, W. Chen, J. Xin, Y. Du, H. Xu, CALPHAD, 41, 2013, 108-118.
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Alloy designation
Mole Fractions
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Table 1. Alloy concentrations for the diffusion couples used in this work.
Ti
A
0.95
0.05
0
B C
0.90 0.80
0 0
0.10 0.20
0.70 0.60 0.53 0.72
0 0 0.02 0.04
0.30 0.40 0.45 0.24
0.87 1.00
0.06 0
0.07 0
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H I
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D E F G
Cu
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Ni
Note: Diffusion couples are configurated as: A-B, A-C, A-D, A-E, I-F, I-G and I-H,
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respectively.
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Table 2. Mobility parameters for Ni, Cu and Ti in fcc Ni-Cu-Ti alloys (all in SI units). Model
Mobility
Parameters Φ
= −287000 − 69.80T [12]
Φ
= −232788 − 71.1T [12]
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Phase
Ni:Va Ni Cu:Va Ni Ti:Va Ni 0 Cu , Ni:Va Ni 0 Ti , Ni:Va Ni 0 Cu ,Ti:Va Ni 0 Ni ,Cu ,Ti:Va Ni 1 Ni ,Cu ,Ti:Va Ni 2 Ni ,Cu ,Ti:Va Ni Cu:Va Cu Ni:Va Cu Ti:Va Cu 0 Cu , Ni:Va Cu 0 Ti , Ni:Va Cu 0 Ni ,Cu ,Ti:Va Cu 1 Ni ,Cu ,Ti:Va Cu 2 Ni ,Cu ,Ti:Va Cu Ti:Va Ti Cu:Va Ti Ni:Va Ti 0 Ni ,Ti:Va Ti 0 Cu , Ni:Va Ti 0 Ni ,Cu ,Ti:Va Ti 1 Ni ,Cu ,Ti:Va Ti 2 Ni ,Cu ,Ti:Va Ti
Φ
= −132849.8 − 81.40T [13]
Φ
= 106790 − 75.4T [12]
Φ
= 351220.4 [13]
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Ni
Φ
= 49316.85
Φ
= 49316.85
Φ
= 49316.85
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Φ
Φ
= −205872 − 82.53T [12]
Φ
= −250125 − 85.3T [12]
Φ
(Ni,Cu,Ti)1(Va)1
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= −132849.8 − 81.40T [13]
Φ
Cu
= 23887 − 17.7T [12]
Φ
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fcc
= 38625.14
Φ
Φ
= 26378.96
Φ
= 26378.96
Φ
= 26378.96
= −132849.8 − 81.40T [13]
Φ
Φ
= −192100 − 84.84T
= −276771.3 − 63.82T [13]
Φ
Ti
= −176117.8 [13]
Φ
8
= 52476.25
= 24365.28
Φ
= 82617.42
Φ
= 82617.42
Φ
= 82617.42
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0.024
0.150
0.029
0.117
0.027
0.074
0.024
0.047
0.031
0.031
0.041
0.050
(10-14) 1.29 1.08 1.10 0.86 0.76 0.84 0.58 0.69 0.57 0.63 0.77 0.82
(10-14) 0.15 0.11 0.14 0.19 0.10 0.08 0.07 0.12 0.08 0.06 0.13 0.11
(10-14) 0.47 0.54 0.42 0.61 0.26 0.19 0.16 0.18 0.12 0.07 0.24 0.31
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Cu
Ni D% CuTi
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Ti
Ni D% TiCu
9
Ni D% CuCu
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Ni D% TiTi
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Concentration
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Table 3. Measured and calculated interdiffusion coefficients at common composition junctions of diffusion couples. (Units: m2/s)
(10-14) 0.28 0.36 0.30 0.24 0.29 0.34 0.28 0.27 0.32 0.38 0.37 0.32
Resources
calculated measured calculated measured calculated Measured calculated measured calculated measured calculated measured
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Cu 0.0 .1
SC
.2
of C
u
.3
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Mo le
Fra
cti on
.4
.5
liq
.6
fc c
bc c
+N i
.8
_B 2
3 Ti
.7
1.0
Ni
0.0
fcc .1
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.9
.2
.3
.4
.5
bcc liq .6
.7
.8
.9
1.0
Ti
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Mole Fraction of Ti
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Fig. 1 Calculated isothermal section of Ni-Cu-Ti ternary system at 1273 K[11].
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Ni
.8
Matano Plane
.6
.4
.2
Cu
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Mole Fractions
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1.0
Ti
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0.0 0
200
400
600
800
1000
Distance (µm)
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Fig. 2 Calculated and experimentally-measured diffusion profiles in Ni0.90Cu0.10/Ni0.95Ti0.05 diffusion couple (A-B) annealed at 1273 K for 150 hrs.
11
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1.0 Ni
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Matano Plane
Mole Fractions
.8
.4
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.6
Cu
.2
Ti 0.0 200
400
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0
600
800
1000
Distance (µm)
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Fig. 3 Calculated and experimentally-measured diffusion profiles in Ni0.95Ti0.05/Ni0.80Cu0.20 diffusion couple (A-C) annealed at 1273 K for 150 hrs.
12
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1.0 Ni
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Matano Plane
.6
.4 Cu
.2
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Mole Fractions
.8
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 4 Calculated and experimentally-measured diffusion profiles in Ni0.70Cu0.30/Ni0.95Ti0.05
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diffusion couple (A-D) annealed at 1273 K for 150 hrs.
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1.0
.2
Matano Plane
Cu .4
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Ni .6
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Mole Fractions
.8
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 5 Calculated and experimentally-measured diffusion profiles in Ni0.60Cu0.40/Ni0.95Ti0.05
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diffusion couple (A-E) annealed at 1273 K for 150 hrs.
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1.0
SC
Matano Plane
.6
.4
.2
Ni
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Mole Fractions
.8
Cu
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 6 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.53Cu0.45Ti0.02
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diffusion couple (I-F) annealed at 1273 K for 150 hrs.
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1.0 Ni
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Matano Plane
.6
.4
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Mole Fractions
.8
Cu
.2
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
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Fig. 7 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.72Cu0.24Ti0.04 diffusion couple (I-G) annealed at 1273 K for 150 hrs.
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1.0 Ni
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Matano Plane
.6
.4
.2
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Mole Fractions
.8
Cu Ti
0.0 200
400
600
800
1000
TE D
0
EP
Distance (µm)
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Fig. 8 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.87Cu0.07Ti0.06 diffusion couple (I-H) annealed at 1273 K for 150 hrs.
17
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.5
.1
1.0 .9
.2
.3
.8 .1
.7
.4
.5
liq
.6
bc c_ B2
fcc +N i3 T i
.08
.10
.1
TE D
.1
.06
.2
bcc
liq
1.0 0.0
.5
.04
Mole Fraction of Ti
.3
.8
Ni
.02
.4
.7
.9
0.0 0.00
.6
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Mo le Fra ctio no fC u
.3
fcc+Ni3Ti
.2
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0.0
Mole Fraction of Cu
Cu
RI PT
.4
.2
.3
.4
.5
.6
.7
0.0 .8
.9
1.0
Ti
EP
Mole Fraction of Ti
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Fig. 9 Calculated and experimentally-measured diffusion paths in Ni-Cu-Ti diffusion couples annealed at 1273 K for 150 hrs.
18
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Atomic mobilities of fcc Ni-Cu-Ti phases were determined. Experimental interdiffusivities were critically evaluated. Main and cross interdiffusivities show their peculiarities. The profiles reveal kinetic importance for alloy microstructures.
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• • • •
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S0925-8388(18)33926-4
DOI:
https://doi.org/10.1016/j.jallcom.2018.10.251
Reference:
JALCOM 48069
To appear in:
Journal of Alloys and Compounds
Received Date: 4 July 2018 Revised Date:
17 October 2018
Accepted Date: 21 October 2018
Please cite this article as: C. Li, S. Huang, Y. Liu, Simulation of atomic mobilities, interdiffusivities and diffusional evolution in fcc Ni-Cu-Ti alloys, Journal of Alloys and Compounds (2018), doi: https:// doi.org/10.1016/j.jallcom.2018.10.251. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Simulation of Atomic Mobilities, Interdiffusivities
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Authors:
SC
RI PT
and Diffusional Evolution in fcc Ni-Cu-Ti Alloys
Chunlin Lia*, Shenming Huangb, Yajun Liub a
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Nanjing Institute of Electronic Technology, Nanjing, Jiangshu, 210039, P. R. China b* School of Materials and Energy, Guangdong University of Technology, Guangzhou, Guangdong, 510006, P. R. China
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* Corresponding Author Tel: 86-10-13851990477
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Email: [email protected], and [email protected]
ACCEPTED MANUSCRIPT
Simulation of Atomic Mobilities, Interdiffusivities and Diffusional Evolution in fcc Ni-Cu-Ti Alloys
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Abstract
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As a contribution to establish a general Ni-based mobility database, the atomic mobilities in fcc Ni-Cu-Ti alloys were explored in this work. Diffusion couples were annealed at 1273 K for 150 hrs, and the interdiffusion coefficients were evaluated at common composition points of diffusion profiles. By combining such interdiffusion coefficients and the available
M AN U
thermodynamic information, the atomic mobilities of Ni, Cu and Ti in fcc Ni-Cu-Ti alloys were assessed as functions of temperature and compositions within the CALPHAD framework. Further verification of the obtained atomic mobilities was conducted by comparing the calculated and measured interdiffusvities, diffusion profiles in diffusion couples and diffusion paths in the Gibbs triangle. Keywords: Mobility; Diffusion; fcc Ni-Cu-Ti alloys; CALPHAD
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1. Introduction
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Characterized by high specific strength, excellent resistance against oxidation and corrosion, Ni-based superalloys are widely used in gas turbine blades and vanes for power plants in high-temperature applications [1-2]. Principally, alloy microstructures can be significantly affected by diffusion process for multicomponent systems, and the knowledge from materials kinetics can be used to optimize alloy performance at high temperatures [3]. In the past decades, studies on alloy kinetics are driven by increasing demands on
AC C
computational simulation of microstructual evolutions, where the phase field approach is one representative technique. Based upon the CALPHAD method, DICTRA was developed to provide fundamental solutions for diffusion problems [4]. DICTRA is capable of simulating diffusion-controlled problems, such as dissolution, solidification, carburization, coarsening and growth. Up to now, commercial thermodynamic databases for titanium alloys are available, and the atomic mobility descriptions for some binary and ternary subsystems have been published. The Ni-Cu-Ti ternary system plays an essential role in exploring Ni-based superalloys. However, related kinetic description based on the CALPHAD framework is still missing. Thus, the objective of this work is to explore accurate mobility description for Ni-Cu-Ti alloys so that a multicomponent Ni-based kinetic database can be established, which indeed provides further insights into diffusion characteristics. 1
ACCEPTED MANUSCRIPT
2. Diffusion Methodology 2.1 Model Description
Mi =
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Based on the absolute reaction rate theory, the atomic mobility parameter of element i can be divided into a frequency term and an activation enthalpy term, which can be expressed as [5]: −Q* + RT ln( M i0 ) Φ 1 1 exp( i )= exp( i ) RT RT RT RT
(1)
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, where M i stands for atomic mobility for element i ; M i0 is the frequency factor; Qi*
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denotes the activation enthalpy; R is the gas constant; T is the absolute temperature. Φ i is equivalent to −Qi* + RT ln( M i0 ) , which can be expanded by the Redlich-Kister polynomial as [6]: Φ i = ∑ x j Φ ij + ∑∑ x j xk [∑ r Φ ij ,k ( x j − xk ) r ] + ∑∑∑ x j xk xl [ν sjkl s Φ ij , k ,l ] j
j
k> j
r
j
(2)
k > j l >k
, where x j is the mole fraction of element i ; Φ ij stands for the mobility end-member of element i to diffuse in j ;
r
Φ ij ,k denotes the interaction parameters for element i to
diffuse in j - k binary systems;
s
Φ ij , k ,l is the ternary interaction parameter for element i ;
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the parameter ν sjkl can be expressed as [6]:
ν sjkl = xs + (1 − xi − x j − xk ) / 3
(3)
According to the Einstein relation, the tracer diffusion coefficients of element i , Di* , can
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be related to its atomic mobility by [7]:
Di* = RTM i
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(4) rN In a number-fixed frame of reference, the interdiffusion flux of element i , J i , can be expressed as: n −1 r J iN = −∑ D% ijn ∇C j
(5)
j =1
, where the superscript N is the number-fixed frame of reference; C j denotes for the ~ volume concentration of element j ; Dijn is the interdiffusion coefficient, which relates the flux of element i to the concentration gradient of element j ; element n is chosen as the dependent element. The interdiffusion coefficient D% n can be expressed as follows [8]: ij
∂µ ∂µ D% ijn = ∑ [(δ ik − xk ) xi M i ( i − i )] ∂x j ∂xn i
(6)
, where the superscript n is the dependent element in the diffusion system, i and j stand 2
ACCEPTED MANUSCRIPT for Cu or Ti, δ ik denotes the Kronecker delta ( δ ik = 0 if i≠k; or δ ik = 1 ); µi is the chemical potential of element i . Provided that the interdiffusion fluxes of all the elements are known, the spatial and temporal evolution of element i can be described by the equation of continuity as follows: ~ ∂Ci + ∇ ⋅ J iN = 0 ∂t
(7)
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, where t is time. DICTRA is a software package based upon the finite difference algorithm, which can be used to solve Eqn. 7 with suitable initial and boundary conditions.
2.2 Evaluation of Interdiffusion Coefficients of element i can be directly evaluated as follows [9]:
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From the concentration profiles developed in diffusion couples, the interdiffusion flux
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r 1 Ci ( x ) J iN = ∫ − + ( x − x0 )dCi 2t Ci or Ci
(8)
, where x0 is the location for the Matano plane. The combination of Eqns. 5 and 8 leads to the following constrains that can be used to extract interdiffusion coefficients at common compositions of diffusion couples:
∫
CCu ( x )
∫
CTi ( x )
CTi−
Ni ∂CCu % Ni ∂CTi ( x − x0 )dCCu = −2t D% CuCu + DCuTi ∂x ∂x
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− + CCu or CCu
or
CTi+
Ni ∂CCu % Ni ∂CTi ( x − x0 )dCTi = −2t D% TiCu + DTiTi ∂x ∂x
(9)
(10)
, where Ni is chosen as the dependent element.
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According to the thermodynamic stability constraints from solid solutions, all the obtained interdiffusion coefficients should be further validated by the following equations
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[10]:
Ni Ni D% CuCu + D% TiTi >0 Ni Ni Ni % Ni D% CuCu D% TiTi − D% CuTi DTiCu ≥ 0 Ni Ni 2 Ni % Ni ( D% CuCu − D% TiTi ) + 4 D% CuTi DTiCu ≥ 0
(11) (12) (13)
3. Experimental Copper (purity: 99.6 wt%), Titanium (purity: 99.9 wt.%), and Nickel (purity: 99.5 wt.%) were used as starting materials to prepared needed alloys. The compositions of such alloys are designed so that the diffusion paths superimposed on the Gibbs triangle can cover a great majority of the fcc phase region. The compositions of designed alloys are listed in Table1, 3
ACCEPTED MANUSCRIPT and the combination manners to form diffusion couples are also given in Table 1. All the alloys were melted in alumina crucibles in an induction melter with a vacuum condition, and the ingots were then homogenized at 1273 K for 150 hrs. After that, the ingots were cut by diamond saw to obtain suitable sized blocks (~10×10×3 mm3), which were then grounded with SiC carbide papers down to 0.5 µm and further polished with diamond slurry (diamond
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size: ~0.25 µm). Seven diffusion couples were prepared by binding the polished surfaces of two blocks together, which were then annealed at 1273 K for 150 hrs in a hot-pressing furnace with an external load of 5 MPa. All the diffusion annealing was carried out under a vacuum atmosphere. In a subsequent step, the annealed diffusion couples were cut parallel to
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the diffusion direction, followed by standard metallographic preparation. The last step is to obtain concentration profiles by means of electron probe microanalysis technique (EPMA) on the polished section.
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4. Results and Discussion
In order to calculate the interdiffusivities by Eqn. 9 and 10, the location of intersection of diffusion couples must be known. Such points were obtained as follows: Firstly, the
TiTi
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experimental concentration curves for diffusion couples were fitted by the highly non-linear equations given in the work of Bouchet et al. [11]. Secondly, such fitted concentration curves were plotted in a Gibbs triangle to locate all the intersection points of diffusion Ni couples. Based upon Eqns. 9 and 10, the two main interdiffusion coefficients, D% CuCu and Ni Ni Ni % % % D , and the two cross interdiffusion coefficients, D and D , can be evaluated at CuTi
TiCu
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the intersections of diffusion paths. Such experimentally-measured interdiffusivities were then used to retrieve the atomic mobilities of Cu, Ti and Ni in fcc Ni-Cu-Ti alloys in the Parrot module in DICTRA. Thermodynamic parameters of fcc Ni-Cu-Ti alloys optimized
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by Zhu et al. [12] are used in the present work, and the calculated phase diagram for 1273 K is presented in Fig. 1. It is noted that the experimentally obtained interdiffusion coefficients are dependent upon thermodynamic information, and the atomic mobilities obtained are independent of thermodynamic factors. Accordingly, such atomic mobilities can be stored in purely kinetic databases in the CALPHAD framework. The obtained interaction parameters and mobility end-members are presented in Table 2, along with those published in the literature [13-14]. Compiled in Table 3 are the calculated and experimentally-measured interdiffusion coefficients at 1273 K, where the agreement is satisfactory. In addition, all the experimentally-measured interdiffusion coefficients obey the stability condition given in Ni Ni Ni Ni Eqns. 11–13. It is obvious that D% CuCu , D% TiTi , D% CuTi and D% TiCu in Table 3 are all Ni Ni Ni % , D% CuTi characterized with positive sign. In addition, DTiTi is generally larger than D% CuCu 4
ACCEPTED MANUSCRIPT Ni and D% TiCu .
The quality of the obtained atomic mobilities can be further validated by simulation diffusion evolution in diffusion couples, when the atomic mobilities are combined with related thermodynamic description. Figs. 2-8 show the experimentally-measured and DICTRA-predicted concentration profiles for Ni-Cu-Ti diffusion couples annealed at 1273 K
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for 150 hrs, in which the Matano plane is superimposed as vertical lines. In general, the calculated results agree well with the corresponding experimentally measured ones, and the concentration curves in such plots generally show a monotonous nature. A comparison between the calculated and experimentally-measured diffusion paths for diffusion couples
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annealed at 1273 K for 150 hrs is presented in Figs. 9. As shown in the figure, all of the diffusion paths developed in the diffusion couples show S-shaped geometry, and there is a good agreement between experimental and predicted calculated diffusion paths.
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5. Conclusions
Based upon the experimentally-measured interdiffusivities and the thermodynamic description for fcc Ni-Cu-Ti alloys, the atomic mobilities of Ni, Cu and Ti for fcc Ni-Cu-Ti alloys were critically assessed in DICTRA. Satisfactory agreement was obtained from comprehensive comparisons made between the calculated and experimental interdiffusivities.
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diffusion couples.
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In order to further validate the quality of the present atomic mobilities, comparison between experimental and model-predicted concentration profiles in diffusion couples was made, and the agreement is satisfactory. In addition, diffusion paths are presented in the Gibbs triangle, which are of great help to gain insights into investigated spatial and temporal evolution in
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References: [1] A. Tamm, A.Aabloo, M. Klintenberg, M. Stocks, A. Caro, Acta Mater., 99, 2015,
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307-312. [2] A. C. Coppa, M. Kappor, R. Noebe, G. B. Thompson, Intermetallics, 67, 2015, 56-62. [3] Y. Nishida, Y. Tsukada, T. Koyama, M. Kurata, J. Nuclear Mater., 466, 2015, 551-559. [4] A. Borgenstam, L. Hoglund, J. Agren, A. Engstrom, J. Phase Equilib. Diff., 21, 2000,
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269-280. [5] J. O. Andersson, J. Ågren, J. Appl. Phys., 72, 1992, 1350-1355. [6] X. He, W. Zhang, M. Yan, C. Chen, Y. Du, L. Zhang, B. Huang, CALPHAD, 49, 2015, 35-40.
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[7] Y. Liu, T. Pan, L. Zhang, D. Yu, Y. Ge, J. Alloys Compd., 476, 2009, 429-435. [8] Z. Zhou, Y. Liu, G. Sheng, F. Lei, Z. Kang, CALPHAD, 48, 2015, 151-156. [9] M. A. Dayananda, J. Phase Equi. Diff., 26, 2005, 441-446. [10] J. S. Kirkaldy, G. R. Purdy, Can. J. Phys., 47, 1969, 865-871. [11] R. Bouchet, R. Mevrel, Acta Mater., 50(19), 2002, 4887-4900. [12] W. J. Zhu, L. I. Duarte. C. Leinenbach, CALPHAD, 47, 2014, 9-22. [13] J. Chen, Y. Liu, F. Lei, G. Sheng, Z. Kang, CALPHAD, 47, 2014, 123-128.
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[14] M. Liu, L. Zhang, W. Chen, J. Xin, Y. Du, H. Xu, CALPHAD, 41, 2013, 108-118.
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Alloy designation
Mole Fractions
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Table 1. Alloy concentrations for the diffusion couples used in this work.
Ti
A
0.95
0.05
0
B C
0.90 0.80
0 0
0.10 0.20
0.70 0.60 0.53 0.72
0 0 0.02 0.04
0.30 0.40 0.45 0.24
0.87 1.00
0.06 0
0.07 0
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H I
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D E F G
Cu
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Ni
Note: Diffusion couples are configurated as: A-B, A-C, A-D, A-E, I-F, I-G and I-H,
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respectively.
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Table 2. Mobility parameters for Ni, Cu and Ti in fcc Ni-Cu-Ti alloys (all in SI units). Model
Mobility
Parameters Φ
= −287000 − 69.80T [12]
Φ
= −232788 − 71.1T [12]
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Phase
Ni:Va Ni Cu:Va Ni Ti:Va Ni 0 Cu , Ni:Va Ni 0 Ti , Ni:Va Ni 0 Cu ,Ti:Va Ni 0 Ni ,Cu ,Ti:Va Ni 1 Ni ,Cu ,Ti:Va Ni 2 Ni ,Cu ,Ti:Va Ni Cu:Va Cu Ni:Va Cu Ti:Va Cu 0 Cu , Ni:Va Cu 0 Ti , Ni:Va Cu 0 Ni ,Cu ,Ti:Va Cu 1 Ni ,Cu ,Ti:Va Cu 2 Ni ,Cu ,Ti:Va Cu Ti:Va Ti Cu:Va Ti Ni:Va Ti 0 Ni ,Ti:Va Ti 0 Cu , Ni:Va Ti 0 Ni ,Cu ,Ti:Va Ti 1 Ni ,Cu ,Ti:Va Ti 2 Ni ,Cu ,Ti:Va Ti
Φ
= −132849.8 − 81.40T [13]
Φ
= 106790 − 75.4T [12]
Φ
= 351220.4 [13]
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Ni
Φ
= 49316.85
Φ
= 49316.85
Φ
= 49316.85
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Φ
Φ
= −205872 − 82.53T [12]
Φ
= −250125 − 85.3T [12]
Φ
(Ni,Cu,Ti)1(Va)1
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= −132849.8 − 81.40T [13]
Φ
Cu
= 23887 − 17.7T [12]
Φ
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fcc
= 38625.14
Φ
Φ
= 26378.96
Φ
= 26378.96
Φ
= 26378.96
= −132849.8 − 81.40T [13]
Φ
Φ
= −192100 − 84.84T
= −276771.3 − 63.82T [13]
Φ
Ti
= −176117.8 [13]
Φ
8
= 52476.25
= 24365.28
Φ
= 82617.42
Φ
= 82617.42
Φ
= 82617.42
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0.024
0.150
0.029
0.117
0.027
0.074
0.024
0.047
0.031
0.031
0.041
0.050
(10-14) 1.29 1.08 1.10 0.86 0.76 0.84 0.58 0.69 0.57 0.63 0.77 0.82
(10-14) 0.15 0.11 0.14 0.19 0.10 0.08 0.07 0.12 0.08 0.06 0.13 0.11
(10-14) 0.47 0.54 0.42 0.61 0.26 0.19 0.16 0.18 0.12 0.07 0.24 0.31
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Cu
Ni D% CuTi
AC C
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Ti
Ni D% TiCu
9
Ni D% CuCu
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Ni D% TiTi
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Concentration
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Table 3. Measured and calculated interdiffusion coefficients at common composition junctions of diffusion couples. (Units: m2/s)
(10-14) 0.28 0.36 0.30 0.24 0.29 0.34 0.28 0.27 0.32 0.38 0.37 0.32
Resources
calculated measured calculated measured calculated Measured calculated measured calculated measured calculated measured
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Cu 0.0 .1
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.2
of C
u
.3
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Mo le
Fra
cti on
.4
.5
liq
.6
fc c
bc c
+N i
.8
_B 2
3 Ti
.7
1.0
Ni
0.0
fcc .1
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.9
.2
.3
.4
.5
bcc liq .6
.7
.8
.9
1.0
Ti
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Mole Fraction of Ti
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Fig. 1 Calculated isothermal section of Ni-Cu-Ti ternary system at 1273 K[11].
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Ni
.8
Matano Plane
.6
.4
.2
Cu
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Mole Fractions
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1.0
Ti
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0.0 0
200
400
600
800
1000
Distance (µm)
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Fig. 2 Calculated and experimentally-measured diffusion profiles in Ni0.90Cu0.10/Ni0.95Ti0.05 diffusion couple (A-B) annealed at 1273 K for 150 hrs.
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1.0 Ni
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Matano Plane
Mole Fractions
.8
.4
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.6
Cu
.2
Ti 0.0 200
400
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0
600
800
1000
Distance (µm)
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Fig. 3 Calculated and experimentally-measured diffusion profiles in Ni0.95Ti0.05/Ni0.80Cu0.20 diffusion couple (A-C) annealed at 1273 K for 150 hrs.
12
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1.0 Ni
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Matano Plane
.6
.4 Cu
.2
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Mole Fractions
.8
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 4 Calculated and experimentally-measured diffusion profiles in Ni0.70Cu0.30/Ni0.95Ti0.05
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diffusion couple (A-D) annealed at 1273 K for 150 hrs.
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1.0
.2
Matano Plane
Cu .4
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Ni .6
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Mole Fractions
.8
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 5 Calculated and experimentally-measured diffusion profiles in Ni0.60Cu0.40/Ni0.95Ti0.05
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diffusion couple (A-E) annealed at 1273 K for 150 hrs.
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1.0
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Matano Plane
.6
.4
.2
Ni
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Mole Fractions
.8
Cu
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
Fig. 6 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.53Cu0.45Ti0.02
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diffusion couple (I-F) annealed at 1273 K for 150 hrs.
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1.0 Ni
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Matano Plane
.6
.4
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Mole Fractions
.8
Cu
.2
Ti
0.0 0
200
400
600
800
1000
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Distance (µm)
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Fig. 7 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.72Cu0.24Ti0.04 diffusion couple (I-G) annealed at 1273 K for 150 hrs.
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1.0 Ni
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Matano Plane
.6
.4
.2
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Mole Fractions
.8
Cu Ti
0.0 200
400
600
800
1000
TE D
0
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Distance (µm)
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Fig. 8 Calculated and experimentally-measured diffusion profiles in Ni/Ni0.87Cu0.07Ti0.06 diffusion couple (I-H) annealed at 1273 K for 150 hrs.
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.5
.1
1.0 .9
.2
.3
.8 .1
.7
.4
.5
liq
.6
bc c_ B2
fcc +N i3 T i
.08
.10
.1
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.1
.06
.2
bcc
liq
1.0 0.0
.5
.04
Mole Fraction of Ti
.3
.8
Ni
.02
.4
.7
.9
0.0 0.00
.6
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Mo le Fra ctio no fC u
.3
fcc+Ni3Ti
.2
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0.0
Mole Fraction of Cu
Cu
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.4
.2
.3
.4
.5
.6
.7
0.0 .8
.9
1.0
Ti
EP
Mole Fraction of Ti
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Fig. 9 Calculated and experimentally-measured diffusion paths in Ni-Cu-Ti diffusion couples annealed at 1273 K for 150 hrs.
18
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Atomic mobilities of fcc Ni-Cu-Ti phases were determined. Experimental interdiffusivities were critically evaluated. Main and cross interdiffusivities show their peculiarities. The profiles reveal kinetic importance for alloy microstructures.
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• • • •
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