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Solute diffusion during high-temperature plastic deformation in alloys
Materials Science and Engineering A 528 (2011) 7196–7199
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Solute diffusion during high-temperature plastic deformation in alloys S.-H. Song ∗ , X.-M. Chen, L.-Q. Weng Department of Materials Science and Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Xili, Shenzhen 518055, China
a r t i c l e
i n f o
Article history: Received 17 November 2010 Received in revised form 28 April 2011 Accepted 7 June 2011 Available online 14 June 2011 Keywords: Deformation Defects Diffusion Alloy
a b s t r a c t Solute volume diffusion during high-temperature plastic deformation in a substitutional solid solution alloy is analyzed theoretically. Both deformation-induced supersaturated vacancy enhanced diffusion effect and dislocation pipe diffusion effect are considered in the model. The model is applied to the prediction of deformation-enhanced phosphorus diffusion in ␥-Fe. Deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion can both enhance the overall phosphorus diffusion coefficient, but the former effect plays a predominant role. At a certain temperature, the deformation-enhanced phosphorus diffusion coefficient is mainly dependent on strain and strain rate, and at each strain rate there is a steady state value for the enhanced diffusion coefficient. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Implementation of many kinetic processes in alloys requires solute diffusion. It is well known that the solute diffusion can be divided into lattice diffusion, pipe diffusion via dislocations, grain boundary diffusion, etc. [1]. Under plastic deformation, the solute atoms within the matrix diffuse mainly through lattice diffusion and pipe diffusion. Lattice diffusion mechanism for substitutional solute atoms is based on vacancy formation and vacancy migration while pipe diffusion is based on the solute atoms associated with the dislocation and transport along the dislocation core. As is well known [2], high-temperature plastic deformation can create excess vacancies, and the supersaturated vacancies should be able to enhance solute diffusion. And dislocation density increases drastically by multiplication during plastic deformation, which can also enhance solute diffusion [3]. In the present work, deformation enhanced solute volume diffusion during high temperature plastic deformation was analyzed theoretically, and the resulting model was applied to predictions of solute diffusion during high temperature plastic deformation in ␥-Fe. 2. Theory During plastic deformation, the vacancy concentration increases with increasing strain until dynamic recrystallization and the supersaturated vacancy concentration (Cv ) is equal to thermal equilibrium vacancy (Cvth ) plus deformation-induced excess vacancy
∗ Corresponding author. Tel.: +86 755 26033465; fax: +86 755 26033504. E-mail addresses: [email protected], [email protected] (S.-H. Song). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.06.015
(Cvex ), i.e., Cv = Cvth + Cvex [4]. The thermal equilibrium vacancy concentration is given by Cvth = B exp(Qvf /kT ) [4], where B is a constant relating to the vibrational entropy of atoms around the vacancy, Qvf is the vacancy formation energy, k is the Boltzmann constant and T is the absolute temperature. During high temperature plastic deformation, excess vacancies may be generated by the motion of mechanical jogs and thermal jogs, and meanwhile the annihilation of the excess vacancies takes place at dislocations and grain boundaries. Consequently, the net ˙ rate of production of excess vacancies with a given strain rate (ε) at a given temperature (T), dCvex /dt, is given by [4] Cj ˝0 dCvex (t)˝0 Dv (t) Dv = Cex − 2 Cex ε˙ + ε˙ − dt 4b3 2 L Qvf
(1)
where is a constant associated with the mechanical production of jogs; is the stress; ˝0 is the atomic volume; b is Burgers vector; Cj is the concentration of thermal jogs, given by exp(− Ej /kT) [4] where Ej is the jog formation energy which may be calculated by 0.1b3 [5] where is the shear modulus; is a parameter which describes the neutralization effect produced by the presence of vacancy emitting and vacancy absorbing jogs and may be calculated by 0.5–10Cj if Cj < 0.05, and else is equal to 0 [4]; Dv is the diffusion coefficient of vacancies; is the dislocation density; is a structural parameter describing the distribution of dislocations, which depends on temperature and can be fitted as 0.75 exp(0.0039T) at temperatures above 700 ◦ C [6], and L is the grain size. Clearly in Eq. (1), the first term on the right side of the equation is related to the vacancy production from mechanical jogs, and the second term from thermal jogs; the third term is related to the vacancy annihilation at dislocations, and the fourth term at grain boundaries.
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
In the plastic deformation before recrystallization at a high temperature, the dislocation density increases by multiplication, and the relationship between dislocation density and time is described by [4,7]
t
(t) = s − (s − 0 ) × exp − =
2 s
b
ˇ
t
≈ s 1 − exp −
ˇ
t
1 − exp −
(2)
ˇ
where s is the steady-state value of dislocation density, 0 is the initial dislocation density before deformation, s is the saturation stress, and ˇ is the characteristic time related to dislocation annihilation and rearrangement during dynamic recovery, which depends on the deformation condition. Since the dislocation density could be down to ∼106 /cm2 in annealed steel but it could be up to ∼1012 /cm2 in heavily deformed steel. Consequently, the initial dislocation density (0 ) may be negligible during high temperature plastic deformation. ˇ is given by [7,8] ˇ=
1 Ad Lnd ε˙ md exp(−Qd /RT )
(3)
where Ad , nd , md are constants, Qd is an apparent activation energy, and R is the gas constant. The stress during high temperature plastic deformation without recrystallization may be expressed as [7,8]
(t) = 0 + b
(t) ≈ b
= s
(t) ≈ b
t
s 1 − exp −
t
1 − exp −
ˇ (4)
ˇ
where 0 is the effective stress which arises mainly from the Peierls–Nabarro force. As shown in Ref. [9], the Peierls–Nabarro force is about 10−5 for close-packed FCC and HCP metals, and is √ much smaller than b s . Therefore this force could be neglected during high temperature plastic deformation. The saturation stress, s , may be modeled using a hyperbolic sine law [4,7]
Q def
ε˙ = A[sinh(as )]n exp −
(5)
RT
where A, a and n are empirical constants, and Qdef is the apparent activation energy for deformation. With Eqs. (2) and (4), Eq. (1) can be rewritten as ˝0 dCvex ˙ s = ε dt Qvf −
t
1 − exp −
Dv Cex s 2
2 2 b2
ˇ
+
t
1 − exp −
ˇ
Cj ˝0 4b3 −
ε˙
Dv Cex L2
(6)
At a constant strain rate and with the boundary condition of Cvex (0) = 0, the solution of Eq. (6) is
t Cvex (t) =
˝0 Qvf
0
× exp
−
x
1 − exp −
˙ s ε
ˇ
+
Cj ˝0 4b3
Dv s2 ˇ t exp − ˇ 2 2 b2
+
Dv (x − t) dx L2
For the lattice diffusion of substitutional solute atoms in a crystal matrix, the diffusion coefficient, Di , can be expressed as [1] Di = ωpv
(8)
where is a material constant; ω is the probability that a solute atom jumps into a vacant nearest-neighbor lattice site, and pv is the probability that any given nearest-neighbor lattice site is vacant. For vacancy mechanism, pv is approximately equal to the vacancy concentration Cv (fraction of vacant lattice sites).ω may be obtained from [1]
G m
ω = f exp −
(9)
kT
where f is the vibrational frequency of solute atoms and Gm is the free energy required for a solute atom to migrate from an equilibrium position to a nearest-neighbor site. As a consequence, Eq. (8) can be rewritten as
G m
Di = fCv exp −
kT
= ı(T )Cv
(10)
During plastic deformation, substitutional solute atoms diffuse in the lattice by vacancy mechanism, so the deformation-induced supersaturated vacancy-enhanced solute diffusion coefficient, Div , can be written as Div = ı(T )(Cvth + Cvex )
(11)
In the absence of deformation, the solute diffusion coefficient is equal to the thermal diffusion coefficient, i.e., Dith , which is usually
given by D0th exp(−(Qth /kT )) [4] where D0th and Qth are the preexponential constant and activation energy for lattice diffusion, respectively, and the vacancy concentration is equal to the thermal equilibrium vacancy concentration. Therefore, Eq. (10) can be rewritten as Dith = ı(T ) × Cvth
(12)
Using Eqs. (11) and (12), one can obtain
Div = Dith
Cvth + Cvex
(13)
Cvth
In addition to supersaturated vacancies, solute diffusion via dislocation pipe can also enhance its overall diffusivity, and the pipe p diffusivity is denoted as Di which is given by [4] p
Di = Dith exp
Q − Q p th kT
(14)
where Qp is the activation energy for pipe diffusion, which is similar to that for grain boundary diffusion. For most metals, the activation energy for grain boundary diffusion, Qgb , is about 0.4–0.6Qth [10]. In the present study, a medium value was chosen, i.e., Qp = Qgb = 0.5Qth . It is demonstrated that the deformation-enhanced solute diffusivity, DiT , may be given by [3,11] p
DiT = Div × (1 − g) + Di × g
(15)
where g is the ratio of solute atoms associated with the dislocation, which can diffuse via the dislocation core. It is well known that the dislocation is a linear defect with an annular stress field around the dislocation line. Normally, the radius of dislocation core is about 1b [12]. Assuming that the cross-section of the dislocation core is annular, the expression of g may be obtained as
ε˙
Dv s2 Dv 2 ˇ x (x − t) + 2 2s 2 exp − 2 2 2 ˇ b b
7197
p
(7)
Eq. (7) describes the excess vacancy concentration as a function of time during high temperature plastic deformation without recrystallization.
g=
b2 Ci Cg
(16)
where Cg is the solute concentration in the matrix which is assumed p as 0.1 at.% in this work and Ci is the solute concentration in the dislocation core. In a dilute atmosphere, the concentration of solute
7198
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
-13
10-10
Diffusion coefficient ( 10
2 -1
Diffusion coefficient (m s )
Pipe diffusion
25
Supersaturated vacancy enhanced diffusion effect
× 20
-11
10
Supersaturated vacancy enhanced diffusion
10-12
10-13
Lattice diffusion
15 10 5
Pipe diffusion effect 0
10-14
Combined effect
30
2 -1
ms )
10-9
0
2
4
6
8
0
2
4
10
Strain (%) Fig. 1. Phosphorus lattice diffusion coefficient, pipe diffusion coefficient and deformation-induced supersaturated vacancy enhanced diffusion coefficient as a function of strain in ␥-Fe at 950 ◦ C (ε˙ = 2.5 × 10−3 s−1 ). p
around the dislocation, ni , which is in units of atoms per unit volume, is given by [13] p
V
ni = ng exp −
kT
(17)
where ng is the average concentration in the matrix in units of atoms per unit volume and V is the interaction energy with the dislocation, which is given by [13] V (r, ) =
4 1 + sin εra3 3 1− r
(18)
where r, are the coordinates relative to the dislocation, and Cottrell and Bilby [13] suggest that the most favorable position where ˚ ε is a constant the solute atoms stay is at = 3/2 and r = 2 A; describing the difference between the radii of solvent and solute atoms, which is given by (ra − rb )/ra where ra and rb are the atomic radii of solvent and solute atoms, respectively; is the slip distance of the dislocation, which is equal to 2ra [13]; and is Poisson’s ratio. With the assumption that the density near the dislocation is close to that in the matrix, Eq. (17) may be rewritten as p
V
Ci ≈ Cg exp −
kT
6
8
10
Strain (%)
(19)
Fig. 2. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of strain in ␥-Fe at 950 ◦ C (ε˙ = 2.5 × 10−3 s−1 ).
much higher than the lattice diffusion coefficient. As described above, the deformation-enhanced diffusion is a combined effect of supersaturated vacancy enhanced diffusion and pipe diffusion. Fig. 2 shows the effect of supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation-enhanced solute diffusion as a function of stain in ␥-Fe at 950 ◦ C with a strain rate of 2.5 × 10−3 s−1 . Obviously, the deformation-induced supersaturated vacancy enhanced diffusion plays a key role in the deformation-enhanced diffusion while the pipe diffusion effect is very small. This is because although the pipe diffusion coefficient is about two orders of magnitude higher than the steady value of the supersaturated vacancy enhanced diffusion coefficient, the ratio of solute atoms associated with the dislocation is too low, which is given by Eq. (16). Fig. 3 represents the temperature dependence of deformation-enhanced diffusion coefficient with the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients plotted for comparison for ␥-Fe after 10% strain at a strain rate of 2.5 × 10−3 s−1 . As seen, the deformation enhanced diffusion coefficient increases drastically with rising temperature above 900 ◦ C. In addition, the supersaturated vacancy enhanced diffusion effect is dominant in the whole temperature range considered. The strain-rate dependence of deformation-enhanced diffusion coefficient is shown in Fig. 4 for ␥-Fe after 10% strain at 950 ◦ C where
With the model described above, one can predict the deformationenhanced volume diffusion coefficient of solute in a binary alloy. 2 -1
ms )
4
Combined effect
-12
Non-equilibrium grain boundary segregation of undesirable elements, e.g. phosphorus, is one of the important factors causing ductility losses of continuously cast products in steel making processes [14]. And the enrichment of solute atoms at grain boundaries is diffusion controlled. Therefore, the deformation-enhanced phosphorus diffusion in ␥-Fe, as an example, has been predicted by the preceding model. Data used in the calculations are listed in Table 1. Fig. 1 shows the phosphorus lattice diffusion coefficient, pipe diffusion coefficient and deformation-induced supersaturated vacancy enhanced diffusion coefficient as a function of strain with a strain rate of 2.5 × 10−3 s−1 at 950 ◦ C in ␥-Fe. Clearly, the lattice diffusion and pipe diffusion coefficients are constant during deformation while the supersaturated vacancy enhanced diffusion coefficient increases with increasing strain until reaching a steady value at a strain of ∼7%. As shown in Fig. 1, the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients are both
Diffusion coefficient ( 10
3. Results and discussion
5
×
3
2
Supersaturated vacancy enhaced diffusion effect
1
0
Pipe diffusion effect
900
920
940
960
980
1000
o
Temperature ( C) Fig. 3. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of temperature in ␥-Fe after 10% strain at 2.5 × 10−3 s−1 .
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
7199
Table 1 Data used in the theoretical calculations.
2 -1
Diffusion coefficient (m s )
(J m−3 ) B ˝0 (m3 ) Dv (m2 s−1 ) b (m) ˛ (MPa−1 ) Qdef (eV) A (s−1 ) n
10
-10
10
-11
10
-12
10
-13
10
-14
10
-15
8.1 × 1010 (1 − 0.91(T − 300)/1810) [4] 0.1 [4] 2.7 [4] 1.21 × 10−29 [4] 3.3 × 10−5 exp(−1.49/kT) [4] 2.58 × 10−10 [4] 1.2 × 10−2 [7] 3.24 [7] 1.56 × 1011 [7] 5.1 [7]
Qd (eV) md Ad nd
Qvf (eV) ra (m) rb (m) Dith (m2 s−1 ) L (m)
0.28 [7] 0.8 [7] 157 [7] −0.1 [8] 0.28 [13] 1.7 [15] 1.27 × 10−10 [16] 1.09 × 10−10 [16] 6.3 × 10−6 exp(−2.01/kT) [17] 70
and dislocation pipe diffusion, can both enhance the overall solute diffusion coefficient, but the supersaturated vacancy-enhanced diffusion plays the leading role. At a given temperature, the deformation-enhanced diffusion coefficient increases with increasing deformation strain until reaching a steady value, and at the same deformation strain it increases with increasing strain rate.
Combined effect
Supersaturated vacancy enhanced diffusion effect
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51071060) and the Science and Technology Foundation of Shenzhen (Grant No. JC200903120199A).
Pipe diffusion effect
References -5
10
-4
10
-3
-2
10
10
-1
10
-1
Strain rate (s ) Fig. 4. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of strain rate in ␥-Fe after 10% strain at 950 ◦ C.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients are also plotted for comparison. Clearly, the overall deformation enhanced diffusion increases with increasing strain rate, and the contribution from supersaturated vacancy enhanced diffusion is predominant.
[12]
4. Conclusions
[13] [14]
The deformation-enhanced solute volume diffusion is analyzed theoretically. Predictions of the deformation-enhanced phosphorus diffusion in ␥-Fe are made. Two effects, which are deformation-induced supersaturated vacancy enhanced diffusion
[10] [11]
[15] [16] [17]
H. Mehrer, Diffusion in Solids, Springer, Berlin, 2007. T. Watanabe, S. Karashima, Phys. Status Solidi 42 (1970) 749. E.W. Hart, Acta Metall. 5 (1957) 597. M. Militzer, W.P. Sun, J.J. Jonas, Acta Metall. Mater. 42 (1994) 133. K. Murty, K. Detemple, O. Kanert, J. Dehosson, Metall. Mater. Trans. A 29 (1998) 153. L.E. Lindgren, K. Domkin, S. Hansson, Mech. Mater. 40 (2008) 907. A. Laasraoui, J.J. Jonas, Metall. Trans. A 22A (1991) 1545. A. Yoshie, H. Morikawa, Y. Onoe, K. Iton, Trans. ISIJ 27 (1987) 425. E. Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press, 2003. Y. Mishin, C. Herzig, Mater. Sci. Eng. A 260 (1999) 55. P.E. Armstrong, W.V. Green, O.D. Sherby, E.G. Zukas, Acta Metall. 21 (1973) 1319. W. Cai, V.V. Bulatov, J.P. Chang, J. Li, S. Yip, Dislocations in Solids, North-Holland, Oxford, 2004. A.H. Cottrell, B.A. Bilby, Proc. Phys. Soc. A 62 (1949) 49. A.M. Guo, Y.H. Wang, D.D. Shen, Z.X. Yuan, S.H. Song, Mater. Sci. Technol. 19 (2003) 1553. S.M. Kim, W.J.L. Buyers, J. Phys. F: Met. Phys. 8 (1978) L103. R.G. Faulkner, S.H. Song, P.E.J. Flewitt, Mater. Sci. Technol. 12 (1996) 904. W.F. Gale, T.C. Totemeier (Eds.), Smithells Metal Reference Book, 8th ed., Butterworth-Heinemann, Oxford, 2004.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Solute diffusion during high-temperature plastic deformation in alloys S.-H. Song ∗ , X.-M. Chen, L.-Q. Weng Department of Materials Science and Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Xili, Shenzhen 518055, China
a r t i c l e
i n f o
Article history: Received 17 November 2010 Received in revised form 28 April 2011 Accepted 7 June 2011 Available online 14 June 2011 Keywords: Deformation Defects Diffusion Alloy
a b s t r a c t Solute volume diffusion during high-temperature plastic deformation in a substitutional solid solution alloy is analyzed theoretically. Both deformation-induced supersaturated vacancy enhanced diffusion effect and dislocation pipe diffusion effect are considered in the model. The model is applied to the prediction of deformation-enhanced phosphorus diffusion in ␥-Fe. Deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion can both enhance the overall phosphorus diffusion coefficient, but the former effect plays a predominant role. At a certain temperature, the deformation-enhanced phosphorus diffusion coefficient is mainly dependent on strain and strain rate, and at each strain rate there is a steady state value for the enhanced diffusion coefficient. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Implementation of many kinetic processes in alloys requires solute diffusion. It is well known that the solute diffusion can be divided into lattice diffusion, pipe diffusion via dislocations, grain boundary diffusion, etc. [1]. Under plastic deformation, the solute atoms within the matrix diffuse mainly through lattice diffusion and pipe diffusion. Lattice diffusion mechanism for substitutional solute atoms is based on vacancy formation and vacancy migration while pipe diffusion is based on the solute atoms associated with the dislocation and transport along the dislocation core. As is well known [2], high-temperature plastic deformation can create excess vacancies, and the supersaturated vacancies should be able to enhance solute diffusion. And dislocation density increases drastically by multiplication during plastic deformation, which can also enhance solute diffusion [3]. In the present work, deformation enhanced solute volume diffusion during high temperature plastic deformation was analyzed theoretically, and the resulting model was applied to predictions of solute diffusion during high temperature plastic deformation in ␥-Fe. 2. Theory During plastic deformation, the vacancy concentration increases with increasing strain until dynamic recrystallization and the supersaturated vacancy concentration (Cv ) is equal to thermal equilibrium vacancy (Cvth ) plus deformation-induced excess vacancy
∗ Corresponding author. Tel.: +86 755 26033465; fax: +86 755 26033504. E-mail addresses: [email protected], [email protected] (S.-H. Song). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.06.015
(Cvex ), i.e., Cv = Cvth + Cvex [4]. The thermal equilibrium vacancy concentration is given by Cvth = B exp(Qvf /kT ) [4], where B is a constant relating to the vibrational entropy of atoms around the vacancy, Qvf is the vacancy formation energy, k is the Boltzmann constant and T is the absolute temperature. During high temperature plastic deformation, excess vacancies may be generated by the motion of mechanical jogs and thermal jogs, and meanwhile the annihilation of the excess vacancies takes place at dislocations and grain boundaries. Consequently, the net ˙ rate of production of excess vacancies with a given strain rate (ε) at a given temperature (T), dCvex /dt, is given by [4] Cj ˝0 dCvex (t)˝0 Dv (t) Dv = Cex − 2 Cex ε˙ + ε˙ − dt 4b3 2 L Qvf
(1)
where is a constant associated with the mechanical production of jogs; is the stress; ˝0 is the atomic volume; b is Burgers vector; Cj is the concentration of thermal jogs, given by exp(− Ej /kT) [4] where Ej is the jog formation energy which may be calculated by 0.1b3 [5] where is the shear modulus; is a parameter which describes the neutralization effect produced by the presence of vacancy emitting and vacancy absorbing jogs and may be calculated by 0.5–10Cj if Cj < 0.05, and else is equal to 0 [4]; Dv is the diffusion coefficient of vacancies; is the dislocation density; is a structural parameter describing the distribution of dislocations, which depends on temperature and can be fitted as 0.75 exp(0.0039T) at temperatures above 700 ◦ C [6], and L is the grain size. Clearly in Eq. (1), the first term on the right side of the equation is related to the vacancy production from mechanical jogs, and the second term from thermal jogs; the third term is related to the vacancy annihilation at dislocations, and the fourth term at grain boundaries.
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
In the plastic deformation before recrystallization at a high temperature, the dislocation density increases by multiplication, and the relationship between dislocation density and time is described by [4,7]
t
(t) = s − (s − 0 ) × exp − =
2 s
b
ˇ
t
≈ s 1 − exp −
ˇ
t
1 − exp −
(2)
ˇ
where s is the steady-state value of dislocation density, 0 is the initial dislocation density before deformation, s is the saturation stress, and ˇ is the characteristic time related to dislocation annihilation and rearrangement during dynamic recovery, which depends on the deformation condition. Since the dislocation density could be down to ∼106 /cm2 in annealed steel but it could be up to ∼1012 /cm2 in heavily deformed steel. Consequently, the initial dislocation density (0 ) may be negligible during high temperature plastic deformation. ˇ is given by [7,8] ˇ=
1 Ad Lnd ε˙ md exp(−Qd /RT )
(3)
where Ad , nd , md are constants, Qd is an apparent activation energy, and R is the gas constant. The stress during high temperature plastic deformation without recrystallization may be expressed as [7,8]
(t) = 0 + b
(t) ≈ b
= s
(t) ≈ b
t
s 1 − exp −
t
1 − exp −
ˇ (4)
ˇ
where 0 is the effective stress which arises mainly from the Peierls–Nabarro force. As shown in Ref. [9], the Peierls–Nabarro force is about 10−5 for close-packed FCC and HCP metals, and is √ much smaller than b s . Therefore this force could be neglected during high temperature plastic deformation. The saturation stress, s , may be modeled using a hyperbolic sine law [4,7]
Q def
ε˙ = A[sinh(as )]n exp −
(5)
RT
where A, a and n are empirical constants, and Qdef is the apparent activation energy for deformation. With Eqs. (2) and (4), Eq. (1) can be rewritten as ˝0 dCvex ˙ s = ε dt Qvf −
t
1 − exp −
Dv Cex s 2
2 2 b2
ˇ
+
t
1 − exp −
ˇ
Cj ˝0 4b3 −
ε˙
Dv Cex L2
(6)
At a constant strain rate and with the boundary condition of Cvex (0) = 0, the solution of Eq. (6) is
t Cvex (t) =
˝0 Qvf
0
× exp
−
x
1 − exp −
˙ s ε
ˇ
+
Cj ˝0 4b3
Dv s2 ˇ t exp − ˇ 2 2 b2
+
Dv (x − t) dx L2
For the lattice diffusion of substitutional solute atoms in a crystal matrix, the diffusion coefficient, Di , can be expressed as [1] Di = ωpv
(8)
where is a material constant; ω is the probability that a solute atom jumps into a vacant nearest-neighbor lattice site, and pv is the probability that any given nearest-neighbor lattice site is vacant. For vacancy mechanism, pv is approximately equal to the vacancy concentration Cv (fraction of vacant lattice sites).ω may be obtained from [1]
G m
ω = f exp −
(9)
kT
where f is the vibrational frequency of solute atoms and Gm is the free energy required for a solute atom to migrate from an equilibrium position to a nearest-neighbor site. As a consequence, Eq. (8) can be rewritten as
G m
Di = fCv exp −
kT
= ı(T )Cv
(10)
During plastic deformation, substitutional solute atoms diffuse in the lattice by vacancy mechanism, so the deformation-induced supersaturated vacancy-enhanced solute diffusion coefficient, Div , can be written as Div = ı(T )(Cvth + Cvex )
(11)
In the absence of deformation, the solute diffusion coefficient is equal to the thermal diffusion coefficient, i.e., Dith , which is usually
given by D0th exp(−(Qth /kT )) [4] where D0th and Qth are the preexponential constant and activation energy for lattice diffusion, respectively, and the vacancy concentration is equal to the thermal equilibrium vacancy concentration. Therefore, Eq. (10) can be rewritten as Dith = ı(T ) × Cvth
(12)
Using Eqs. (11) and (12), one can obtain
Div = Dith
Cvth + Cvex
(13)
Cvth
In addition to supersaturated vacancies, solute diffusion via dislocation pipe can also enhance its overall diffusivity, and the pipe p diffusivity is denoted as Di which is given by [4] p
Di = Dith exp
Q − Q p th kT
(14)
where Qp is the activation energy for pipe diffusion, which is similar to that for grain boundary diffusion. For most metals, the activation energy for grain boundary diffusion, Qgb , is about 0.4–0.6Qth [10]. In the present study, a medium value was chosen, i.e., Qp = Qgb = 0.5Qth . It is demonstrated that the deformation-enhanced solute diffusivity, DiT , may be given by [3,11] p
DiT = Div × (1 − g) + Di × g
(15)
where g is the ratio of solute atoms associated with the dislocation, which can diffuse via the dislocation core. It is well known that the dislocation is a linear defect with an annular stress field around the dislocation line. Normally, the radius of dislocation core is about 1b [12]. Assuming that the cross-section of the dislocation core is annular, the expression of g may be obtained as
ε˙
Dv s2 Dv 2 ˇ x (x − t) + 2 2s 2 exp − 2 2 2 ˇ b b
7197
p
(7)
Eq. (7) describes the excess vacancy concentration as a function of time during high temperature plastic deformation without recrystallization.
g=
b2 Ci Cg
(16)
where Cg is the solute concentration in the matrix which is assumed p as 0.1 at.% in this work and Ci is the solute concentration in the dislocation core. In a dilute atmosphere, the concentration of solute
7198
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
-13
10-10
Diffusion coefficient ( 10
2 -1
Diffusion coefficient (m s )
Pipe diffusion
25
Supersaturated vacancy enhanced diffusion effect
× 20
-11
10
Supersaturated vacancy enhanced diffusion
10-12
10-13
Lattice diffusion
15 10 5
Pipe diffusion effect 0
10-14
Combined effect
30
2 -1
ms )
10-9
0
2
4
6
8
0
2
4
10
Strain (%) Fig. 1. Phosphorus lattice diffusion coefficient, pipe diffusion coefficient and deformation-induced supersaturated vacancy enhanced diffusion coefficient as a function of strain in ␥-Fe at 950 ◦ C (ε˙ = 2.5 × 10−3 s−1 ). p
around the dislocation, ni , which is in units of atoms per unit volume, is given by [13] p
V
ni = ng exp −
kT
(17)
where ng is the average concentration in the matrix in units of atoms per unit volume and V is the interaction energy with the dislocation, which is given by [13] V (r, ) =
4 1 + sin εra3 3 1− r
(18)
where r, are the coordinates relative to the dislocation, and Cottrell and Bilby [13] suggest that the most favorable position where ˚ ε is a constant the solute atoms stay is at = 3/2 and r = 2 A; describing the difference between the radii of solvent and solute atoms, which is given by (ra − rb )/ra where ra and rb are the atomic radii of solvent and solute atoms, respectively; is the slip distance of the dislocation, which is equal to 2ra [13]; and is Poisson’s ratio. With the assumption that the density near the dislocation is close to that in the matrix, Eq. (17) may be rewritten as p
V
Ci ≈ Cg exp −
kT
6
8
10
Strain (%)
(19)
Fig. 2. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of strain in ␥-Fe at 950 ◦ C (ε˙ = 2.5 × 10−3 s−1 ).
much higher than the lattice diffusion coefficient. As described above, the deformation-enhanced diffusion is a combined effect of supersaturated vacancy enhanced diffusion and pipe diffusion. Fig. 2 shows the effect of supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation-enhanced solute diffusion as a function of stain in ␥-Fe at 950 ◦ C with a strain rate of 2.5 × 10−3 s−1 . Obviously, the deformation-induced supersaturated vacancy enhanced diffusion plays a key role in the deformation-enhanced diffusion while the pipe diffusion effect is very small. This is because although the pipe diffusion coefficient is about two orders of magnitude higher than the steady value of the supersaturated vacancy enhanced diffusion coefficient, the ratio of solute atoms associated with the dislocation is too low, which is given by Eq. (16). Fig. 3 represents the temperature dependence of deformation-enhanced diffusion coefficient with the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients plotted for comparison for ␥-Fe after 10% strain at a strain rate of 2.5 × 10−3 s−1 . As seen, the deformation enhanced diffusion coefficient increases drastically with rising temperature above 900 ◦ C. In addition, the supersaturated vacancy enhanced diffusion effect is dominant in the whole temperature range considered. The strain-rate dependence of deformation-enhanced diffusion coefficient is shown in Fig. 4 for ␥-Fe after 10% strain at 950 ◦ C where
With the model described above, one can predict the deformationenhanced volume diffusion coefficient of solute in a binary alloy. 2 -1
ms )
4
Combined effect
-12
Non-equilibrium grain boundary segregation of undesirable elements, e.g. phosphorus, is one of the important factors causing ductility losses of continuously cast products in steel making processes [14]. And the enrichment of solute atoms at grain boundaries is diffusion controlled. Therefore, the deformation-enhanced phosphorus diffusion in ␥-Fe, as an example, has been predicted by the preceding model. Data used in the calculations are listed in Table 1. Fig. 1 shows the phosphorus lattice diffusion coefficient, pipe diffusion coefficient and deformation-induced supersaturated vacancy enhanced diffusion coefficient as a function of strain with a strain rate of 2.5 × 10−3 s−1 at 950 ◦ C in ␥-Fe. Clearly, the lattice diffusion and pipe diffusion coefficients are constant during deformation while the supersaturated vacancy enhanced diffusion coefficient increases with increasing strain until reaching a steady value at a strain of ∼7%. As shown in Fig. 1, the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients are both
Diffusion coefficient ( 10
3. Results and discussion
5
×
3
2
Supersaturated vacancy enhaced diffusion effect
1
0
Pipe diffusion effect
900
920
940
960
980
1000
o
Temperature ( C) Fig. 3. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of temperature in ␥-Fe after 10% strain at 2.5 × 10−3 s−1 .
S.-H. Song et al. / Materials Science and Engineering A 528 (2011) 7196–7199
7199
Table 1 Data used in the theoretical calculations.
2 -1
Diffusion coefficient (m s )
(J m−3 ) B ˝0 (m3 ) Dv (m2 s−1 ) b (m) ˛ (MPa−1 ) Qdef (eV) A (s−1 ) n
10
-10
10
-11
10
-12
10
-13
10
-14
10
-15
8.1 × 1010 (1 − 0.91(T − 300)/1810) [4] 0.1 [4] 2.7 [4] 1.21 × 10−29 [4] 3.3 × 10−5 exp(−1.49/kT) [4] 2.58 × 10−10 [4] 1.2 × 10−2 [7] 3.24 [7] 1.56 × 1011 [7] 5.1 [7]
Qd (eV) md Ad nd
Qvf (eV) ra (m) rb (m) Dith (m2 s−1 ) L (m)
0.28 [7] 0.8 [7] 157 [7] −0.1 [8] 0.28 [13] 1.7 [15] 1.27 × 10−10 [16] 1.09 × 10−10 [16] 6.3 × 10−6 exp(−2.01/kT) [17] 70
and dislocation pipe diffusion, can both enhance the overall solute diffusion coefficient, but the supersaturated vacancy-enhanced diffusion plays the leading role. At a given temperature, the deformation-enhanced diffusion coefficient increases with increasing deformation strain until reaching a steady value, and at the same deformation strain it increases with increasing strain rate.
Combined effect
Supersaturated vacancy enhanced diffusion effect
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51071060) and the Science and Technology Foundation of Shenzhen (Grant No. JC200903120199A).
Pipe diffusion effect
References -5
10
-4
10
-3
-2
10
10
-1
10
-1
Strain rate (s ) Fig. 4. Effect of deformation-induced supersaturated vacancy enhanced diffusion and pipe diffusion on the overall deformation enhanced diffusion of phosphorus as a function of strain rate in ␥-Fe after 10% strain at 950 ◦ C.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
the supersaturated vacancy enhanced diffusion and pipe diffusion coefficients are also plotted for comparison. Clearly, the overall deformation enhanced diffusion increases with increasing strain rate, and the contribution from supersaturated vacancy enhanced diffusion is predominant.
[12]
4. Conclusions
[13] [14]
The deformation-enhanced solute volume diffusion is analyzed theoretically. Predictions of the deformation-enhanced phosphorus diffusion in ␥-Fe are made. Two effects, which are deformation-induced supersaturated vacancy enhanced diffusion
[10] [11]
[15] [16] [17]
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