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Reply to Comments on “Interdiffusion in the β phase region of the Ni–Al–Cr system”
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Scripta Materialia 62 (2010) 632–634 www.elsevier.com/locate/scriptamat
Reply to Comments on “Interdiffusion in the b phase region of the Ni–Al–Cr system” H. Wei* and X.F. Sun Superalloys Division, Institute of Metal Research, CAS, Shenyang 110016, China Received 15 January 2010; accepted 15 January 2010 Available online 21 January 2010
A recent comment on a previously published paper addressed the invalid explanation of the off-diagonal interdiffusion coefficients of the b-Ni(Al,Cr) phase in the Ni–Cr–Al system according to the symmetric property of the thermodynamic matrix. In this paper, the experimental data presented by Hou et al. was reanalyzed and the interdiffusion coefficient matrix was estimated again. The results of the analysis have been discussed in terms of a brief reply to the comments of Liu and Liang. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; Intermetallic compound; Ni–Cr–Al
In a previously published paper [1] we reported the interdiffusion coefficient matrix of the b-Ni(Al,Cr) phase in a Ni–Cr–Al system. According to a comment by Liu and Liang [2], it is necessary to first check the reliability of the data present in Figure 4 in Hou et al. [1]. Thus all experimental data were analyzed again, and then the interdiffusion coefficient matrix of the bNi(Al,Cr) phase under investigation estimated using the same method as Hou et al. [1]. The experimentally obtained concentration profiles were normalized with respect to the Boltzmann variable and then presented in the left-hand side of Figure 1. The shapes of the curves were expected to nearly resemble each other. The corresponding interdiffusion fluxes of the components were estimated using Eq. (5) in Hou et al. [1] on the basis of the experimentally obtained concentration profiles and presented in the right-hand side of Figure 1. The interdiffusion flux of Al at each temperature was observed to be positive, while the interdiffusion fluxes of Ni and Cr were negative. The interdiffusion flux of Cr was expected to be less than those of Ni and Al, possibly resulting from the flatness of its concentration profile. Uphill diffusion among Ni, Co and Al components could not be found. This means that Ni, Al and Cr diffuse down their concentration gradients over the diffusion zone examined. We subsequently estimated the interdiffusion coefficient matrix using the method described in Hou et al. [1]. The obtained elements in the * Corresponding author. E-mail: [email protected]
matrix are given in Table 1. We can see from Table 1 that the off-diagonal interdiffusion coefficients possess the same signs as the diagonal interdiffusion coefficients. These results can be explained according to the symmetric property of the thermodynamic matrix, as discussed in Liu and Liang [2]. Figure 2 clearly reveals that the diffusion potentials (lCr lNi) and (lAl lNi) change in the same direction as the variables xAl and xCr in the region of stable b phase in the Ni–Cr–Al system, thereby indicating the symmetric property of the thermodynamic matrix (Eq. (3) in Liu and Liang [2]). It is consistent with the results in Table 1, implying that there is something wrong with the data in Figure 4 of Hou et al. [1]. Using the same concentration profiles, we carefully examined the previous program used to solve the set of four equations of types (2) and (3) in Hou et al. [1] and the details of the calculations by comparison with the present results and with those obtained using the MultiDiflux program developed by Dayananda and RamMohan (https://engineering.purdue.edu/MSE/Research/ MultiDiFlux/index.html). A possible reason leading to the mistake made in Hou et al. [1] is that inappropriate boundary conditions were used when calculating the interdiffusion flux of the components using the relation: Z x þ ðC i Ci Þ ~ J i ðx Þ ¼ Y i ð1 Y i Þdx 2t 1 Z þ1 þ ð1 Y i Þ Y i dx ði ¼ Al; Cr; NiÞ x
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.01.030
β /α
1123K
intersecting diffusion paths
0.06
0.00 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
β /α
β
intersecting diffusion paths
0.06
0.02
1123K for 8h
~
JCr ~
JNi β /α
15
1223K for 8h
~
JAl
10 5 0
~
JCr
-5
~
JNi
-10 -15 30
β /α
β
~
0.04
1323K
intersecting diffusion paths
0.06 Al / 5h Cr / 5h Al / 8h Cr / 8h
0.04 0.02
β /α
1323K for 8h
7
Atomic fraction
β /α
~
JAl
7
Al / 5h Cr / 5h Al / 8h Cr / 8h
0.00 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
70 60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 20
633
1223K
~
Atomic fraction
0.02
~
0.04
6
Al / 5h Cr / 5h Al / 8h Cr / 8h
Ji / 10 atom fraction-m/s
β
Ji (10 atom fraction-m/s)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
Ji / 10 atom fraction-m/s
Atomic fraction
H. Wei, X. F. Sun / Scripta Materialia 62 (2010) 632–634
20
~
JAl
10 0
~
-10 ~
JCr
JNI
-20 -30
0.00 0
1
2
3
4
5 -7
6
7
8
0.0
9
0.5
1.0
0.5
1.5
2.0
2.5 -7
λ / 10 m/s
3.0
3.5
4.0
4.5
0.5
λ / 10 m/s
Figure 1. Distance–concentration plots and the corresponding profiles of interdiffusion fluxes calculated in only the b single phase region of the Ni– Al–Cr system. The solid lines on the left-hand side are the fitted concentration profiles by cubic splines. Table 1. Elements in the interdiffusion coefficient matrix of the b phase under investigation. Composition (mole fraction)
1123 K
1223 K
1323 K
~ ij (1015 m2 s1) D
CAl
CCr
~ AlAl D
~ AlCr D
~ CrAl D
~ CrCr D
0.5338 0.5327 0.5318 0.5301 0.5298 0.5280 0.5277 0.5272 0.5258 0.5201 0.5170 0.5453 0.5434 0.5423 0.5382 0.5373 0.5354 0.5345 0.5305 0.5253 0.5284 0.5221
0.0196 0.0206 0.0220 0.0226 0.0230 0.0165 0.0171 0.0183 0.0189 0.0209 0.0220 0.0196 0.0198 0.0211 0.0227 0.0237 0.0254 0.0271 0.0275 0.0298 0.0276 0.0309
6.64 6.31 5.96 5.19 4.95 9.67 9.59 9.44 8.86 7.21 6.31 23.98 23.68 23.09 22.61 22.01 20.55 19.78 18.58 16.54 15.75 14.07
0.182 0.198 0.200 0.220 0.221 0.425 0.436 0.448 0.502 0.527 0.542 0.545 0.508 0.562 0.643 0.659 0.764 0.816 0.807 0.798 0.754 0.851
0.190 0.156 0.180 0.146 0.196 1.151 1.072 1.054 1.141 1.105 0.951 2.102 2.150 2.144 2.222 2.144 2.106 1.904 2.124 1.890 2.014 1.958
1.85 1.95 2.04 2.07 2.13 2.41 2.44 2.45 2.51 2.65 2.71 3.33 3.41 3.43 3.52 3.58 3.55 3.57 3.68 3.67 3.70 3.77
The compositions quoted correspond to those at which the diffusion paths from the couples intersected.
634
H. Wei, X. F. Sun / Scripta Materialia 62 (2010) 632–634 60
(a)
μ(Al)-μ(Ni)
55
x(Al)=0.528
50
0.523 0.520
45 0.517 40
10
3
35 0 5 -3 10
10 15 20 25 30 35 40 45 50 Mole fraction Cr
10
(b) 9 x(Cr)=0.05
μ(Cr)-μ(Ni)
8
0.01 0.005
7 0.03 6 5
0.02
where C i is the concentration of component i, C i and Cþ i are the concentrations of component i at each end of the diffusion couple, x is distance and þ Y i ¼ ðC i C i Þ=ðC i C i Þ. It is clear from Figure 2 in Hou et al. [1] that the concentration profiles orderly span the b-NiAl, b-NiAl + a-Cr, b-NiAl, b-NiAl + c0 Ni3Al, c0 -Ni3Al and c-Ni + c0 -Ni3Al phase fields. This indicates that the length of the nearly horizontal portion of the curves basically corresponds to the b-NiAl single phase region, however, possibly including the partial two phase region b-NiAl + a-Cr, because it is hard to distinguish the b-NiAl/a-Cr interface from concentration profiles alone. Because the above equation is only valid in the single phase field, inappropriate boundary conditions have unintentionally caused an error when calculating the interdiffusion flux associated with the interdiffusion coefficient matrix. The criticism of the previous paper by Liu and Liang is reasonable. On the basis of the previous experimental data the interdiffusion coefficient matrix of the b phase region of the Ni–Al–Cr system have been re-estimated and are reported in this reply.
4 3 10 4 2 0.45
0.50 0.55 Mole fraction Al
0.60
Figure 2. (li lNi) vs. lj plots for (a) i = Al, j = Cr and (b) i = Cr, j = Al in the b phase of the Ni–Al–Cr system at 1223 K, calculated using Thermo-Calc software on the basis of the TTNi7 database.
[1] G.C. Hou, H. Wei, N.R. Zhao, X.F. Sun, H.R. Guan, Z.Q. Hu, Scr. Mater. 58 (2008) 57–60. [2] Y. Liu, D. Liang, Scr. Mater. 62 (2010) 629–631.
Scripta Materialia 62 (2010) 632–634 www.elsevier.com/locate/scriptamat
Reply to Comments on “Interdiffusion in the b phase region of the Ni–Al–Cr system” H. Wei* and X.F. Sun Superalloys Division, Institute of Metal Research, CAS, Shenyang 110016, China Received 15 January 2010; accepted 15 January 2010 Available online 21 January 2010
A recent comment on a previously published paper addressed the invalid explanation of the off-diagonal interdiffusion coefficients of the b-Ni(Al,Cr) phase in the Ni–Cr–Al system according to the symmetric property of the thermodynamic matrix. In this paper, the experimental data presented by Hou et al. was reanalyzed and the interdiffusion coefficient matrix was estimated again. The results of the analysis have been discussed in terms of a brief reply to the comments of Liu and Liang. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; Intermetallic compound; Ni–Cr–Al
In a previously published paper [1] we reported the interdiffusion coefficient matrix of the b-Ni(Al,Cr) phase in a Ni–Cr–Al system. According to a comment by Liu and Liang [2], it is necessary to first check the reliability of the data present in Figure 4 in Hou et al. [1]. Thus all experimental data were analyzed again, and then the interdiffusion coefficient matrix of the bNi(Al,Cr) phase under investigation estimated using the same method as Hou et al. [1]. The experimentally obtained concentration profiles were normalized with respect to the Boltzmann variable and then presented in the left-hand side of Figure 1. The shapes of the curves were expected to nearly resemble each other. The corresponding interdiffusion fluxes of the components were estimated using Eq. (5) in Hou et al. [1] on the basis of the experimentally obtained concentration profiles and presented in the right-hand side of Figure 1. The interdiffusion flux of Al at each temperature was observed to be positive, while the interdiffusion fluxes of Ni and Cr were negative. The interdiffusion flux of Cr was expected to be less than those of Ni and Al, possibly resulting from the flatness of its concentration profile. Uphill diffusion among Ni, Co and Al components could not be found. This means that Ni, Al and Cr diffuse down their concentration gradients over the diffusion zone examined. We subsequently estimated the interdiffusion coefficient matrix using the method described in Hou et al. [1]. The obtained elements in the * Corresponding author. E-mail: [email protected]
matrix are given in Table 1. We can see from Table 1 that the off-diagonal interdiffusion coefficients possess the same signs as the diagonal interdiffusion coefficients. These results can be explained according to the symmetric property of the thermodynamic matrix, as discussed in Liu and Liang [2]. Figure 2 clearly reveals that the diffusion potentials (lCr lNi) and (lAl lNi) change in the same direction as the variables xAl and xCr in the region of stable b phase in the Ni–Cr–Al system, thereby indicating the symmetric property of the thermodynamic matrix (Eq. (3) in Liu and Liang [2]). It is consistent with the results in Table 1, implying that there is something wrong with the data in Figure 4 of Hou et al. [1]. Using the same concentration profiles, we carefully examined the previous program used to solve the set of four equations of types (2) and (3) in Hou et al. [1] and the details of the calculations by comparison with the present results and with those obtained using the MultiDiflux program developed by Dayananda and RamMohan (https://engineering.purdue.edu/MSE/Research/ MultiDiFlux/index.html). A possible reason leading to the mistake made in Hou et al. [1] is that inappropriate boundary conditions were used when calculating the interdiffusion flux of the components using the relation: Z x þ ðC i Ci Þ ~ J i ðx Þ ¼ Y i ð1 Y i Þdx 2t 1 Z þ1 þ ð1 Y i Þ Y i dx ði ¼ Al; Cr; NiÞ x
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.01.030
β /α
1123K
intersecting diffusion paths
0.06
0.00 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
β /α
β
intersecting diffusion paths
0.06
0.02
1123K for 8h
~
JCr ~
JNi β /α
15
1223K for 8h
~
JAl
10 5 0
~
JCr
-5
~
JNi
-10 -15 30
β /α
β
~
0.04
1323K
intersecting diffusion paths
0.06 Al / 5h Cr / 5h Al / 8h Cr / 8h
0.04 0.02
β /α
1323K for 8h
7
Atomic fraction
β /α
~
JAl
7
Al / 5h Cr / 5h Al / 8h Cr / 8h
0.00 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
70 60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 20
633
1223K
~
Atomic fraction
0.02
~
0.04
6
Al / 5h Cr / 5h Al / 8h Cr / 8h
Ji / 10 atom fraction-m/s
β
Ji (10 atom fraction-m/s)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08
Ji / 10 atom fraction-m/s
Atomic fraction
H. Wei, X. F. Sun / Scripta Materialia 62 (2010) 632–634
20
~
JAl
10 0
~
-10 ~
JCr
JNI
-20 -30
0.00 0
1
2
3
4
5 -7
6
7
8
0.0
9
0.5
1.0
0.5
1.5
2.0
2.5 -7
λ / 10 m/s
3.0
3.5
4.0
4.5
0.5
λ / 10 m/s
Figure 1. Distance–concentration plots and the corresponding profiles of interdiffusion fluxes calculated in only the b single phase region of the Ni– Al–Cr system. The solid lines on the left-hand side are the fitted concentration profiles by cubic splines. Table 1. Elements in the interdiffusion coefficient matrix of the b phase under investigation. Composition (mole fraction)
1123 K
1223 K
1323 K
~ ij (1015 m2 s1) D
CAl
CCr
~ AlAl D
~ AlCr D
~ CrAl D
~ CrCr D
0.5338 0.5327 0.5318 0.5301 0.5298 0.5280 0.5277 0.5272 0.5258 0.5201 0.5170 0.5453 0.5434 0.5423 0.5382 0.5373 0.5354 0.5345 0.5305 0.5253 0.5284 0.5221
0.0196 0.0206 0.0220 0.0226 0.0230 0.0165 0.0171 0.0183 0.0189 0.0209 0.0220 0.0196 0.0198 0.0211 0.0227 0.0237 0.0254 0.0271 0.0275 0.0298 0.0276 0.0309
6.64 6.31 5.96 5.19 4.95 9.67 9.59 9.44 8.86 7.21 6.31 23.98 23.68 23.09 22.61 22.01 20.55 19.78 18.58 16.54 15.75 14.07
0.182 0.198 0.200 0.220 0.221 0.425 0.436 0.448 0.502 0.527 0.542 0.545 0.508 0.562 0.643 0.659 0.764 0.816 0.807 0.798 0.754 0.851
0.190 0.156 0.180 0.146 0.196 1.151 1.072 1.054 1.141 1.105 0.951 2.102 2.150 2.144 2.222 2.144 2.106 1.904 2.124 1.890 2.014 1.958
1.85 1.95 2.04 2.07 2.13 2.41 2.44 2.45 2.51 2.65 2.71 3.33 3.41 3.43 3.52 3.58 3.55 3.57 3.68 3.67 3.70 3.77
The compositions quoted correspond to those at which the diffusion paths from the couples intersected.
634
H. Wei, X. F. Sun / Scripta Materialia 62 (2010) 632–634 60
(a)
μ(Al)-μ(Ni)
55
x(Al)=0.528
50
0.523 0.520
45 0.517 40
10
3
35 0 5 -3 10
10 15 20 25 30 35 40 45 50 Mole fraction Cr
10
(b) 9 x(Cr)=0.05
μ(Cr)-μ(Ni)
8
0.01 0.005
7 0.03 6 5
0.02
where C i is the concentration of component i, C i and Cþ i are the concentrations of component i at each end of the diffusion couple, x is distance and þ Y i ¼ ðC i C i Þ=ðC i C i Þ. It is clear from Figure 2 in Hou et al. [1] that the concentration profiles orderly span the b-NiAl, b-NiAl + a-Cr, b-NiAl, b-NiAl + c0 Ni3Al, c0 -Ni3Al and c-Ni + c0 -Ni3Al phase fields. This indicates that the length of the nearly horizontal portion of the curves basically corresponds to the b-NiAl single phase region, however, possibly including the partial two phase region b-NiAl + a-Cr, because it is hard to distinguish the b-NiAl/a-Cr interface from concentration profiles alone. Because the above equation is only valid in the single phase field, inappropriate boundary conditions have unintentionally caused an error when calculating the interdiffusion flux associated with the interdiffusion coefficient matrix. The criticism of the previous paper by Liu and Liang is reasonable. On the basis of the previous experimental data the interdiffusion coefficient matrix of the b phase region of the Ni–Al–Cr system have been re-estimated and are reported in this reply.
4 3 10 4 2 0.45
0.50 0.55 Mole fraction Al
0.60
Figure 2. (li lNi) vs. lj plots for (a) i = Al, j = Cr and (b) i = Cr, j = Al in the b phase of the Ni–Al–Cr system at 1223 K, calculated using Thermo-Calc software on the basis of the TTNi7 database.
[1] G.C. Hou, H. Wei, N.R. Zhao, X.F. Sun, H.R. Guan, Z.Q. Hu, Scr. Mater. 58 (2008) 57–60. [2] Y. Liu, D. Liang, Scr. Mater. 62 (2010) 629–631.