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Mechanisms of volume diffusion in metals near the Debye temperature
Materials Chemistry and Physics 219 (2018) 273–277
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Mechanisms of volume diffusion in metals near the Debye temperature E.S. Smirnova, V.N. Chuvil'deev, A.V. Nokhrin
∗
T
Lobachevsky State University of Nizhny Novgorod, 603950, Nizhny Novgorod, Russia
H I GH L IG H T S
model is proposed at temperatures above (T > T ) and below (T < T ) the Debye temperature. • AAtdiffusion can be described in localized melting terms. • At TT >< TT diffusion occurs as a result of fluctuation formation of a hollow diffusion corridor. • The effect ofdiffusion reduced diffusion activation energy at low temperatures explained. • D
D
D D
A R T I C LE I N FO
A B S T R A C T
Keywords: Diffusion Metals and alloys Crystal structure Debye temperature
The paper offers a phenomenological model of volume self-diffusion and interstitial diffusion at high (T > TD) and low (T < TD) temperatures (where TD stands for the Debye temperature). Diffusion mechanisms at high and low temperatures were shown to differ greatly. Diffusion at high temperatures occurs as a result of fluctuations that can be described in the localized melting terms – 'liquid diffusion corridor' formation. At low temperatures when melting is difficult for a number of reasons, diffusion occurs through a 'hollow diffusion corridor' formed by fluctuation. Activation energy calculations for self-diffusion agree well with the T > TD experiment and show a dramatic increase in the activation energy at T < TD. Interstitial diffusion activation energy calculated for BCC metals agrees well with the experiment of the whole temperature range and helps to explain why diffusion activation energy goes down at low temperatures.
1. Introduction To date, experimental data on volume self-diffusion in various metal systems at high temperatures T ≥ 0.5Tm (Tm stands for material melting temperature) is extensive [1–11]. Experimental values of volume selfdiffusion activation energy Qvs that are generally obtained while studying diffusion permeability at 0.4–0.9Tm lie within the 18–20 kTm range [1–4]. A vacancy mechanism is traditionally used to provide a theoretical description of volume self-diffusion and interstitial diffusion. According to the classical concept of diffusion mass transfer mechanisms observed in the crystal lattice, a diffusing atom shall jump into the vacancy formed. This concept brings the following volume self-diffusion coefficient expression [2–4]: Dvs = Dvs0 exp(-(Q1s + Q2s)/kT)
(1)
Dvs0 = fza ν0exp((S1s + S2s))/k
(2)
2
where Dvs0 is the pre-exponential factor; f ∼ 1 is the correlation factor; ∗
z is the coordination number; ν0 is the frequency of atomic vibrations; Q1s and S1s are the energy and entropy of vacancy formation; Q2s and S2s are the energy and entropy of vacancy migration. Unlike volume self-diffusion, interstitial diffusion in BCC metals is experimentally studied over a wider temperature range: apart from Т≥0.4–0.5Tm temperatures generally used to study diffusion, interstitial diffusion was studied at lower temperatures (∼0.2Tm). As a rule, indirect methods are applied to this end. One of those methods is based on studying principles of internal friction [12,13]. It is traditionally assumed that interstitial diffusion is the reason for two internal friction peaks in BCC metals: Snoek peak at temperatures close to 0.2Tm and Snoek-Koster peak at 0.3–0.4Tm temperatures [12,13]. Standard methods help to identify interstitial diffusion activation energy and in some cases, the concentration of impurity atoms in a solid solution [1–4]. However, it should be emphasized that the values of diffusion activation energies in atoms of one and the same interstitial impurity obtained for those temperature ranges differ greatly [13]. Thus, when studying Snoek relaxation for carbon atoms in iron [13], the values of interstitial diffusion activation energy were obtained within the range
Corresponding author. E-mail address: [email protected] (A.V. Nokhrin).
https://doi.org/10.1016/j.matchemphys.2018.08.047 Received 11 November 2017; Received in revised form 9 June 2018; Accepted 19 August 2018 Available online 20 August 2018 0254-0584/ © 2018 Elsevier B.V. All rights reserved.
Materials Chemistry and Physics 219 (2018) 273–277
E.S. Smirnova et al.
of Qvi∼5.1–5.5 kTm, where as the Snoek-Koster relaxation values stood higher at Qvi∼8–9 kTm. An interstitial mechanism is used to provide a theoretical description for interstitial diffusion [2–4]. It is assumed that interstitial atoms need no vacancies for a diffusion jump as they spend energy only on migration from one interstitial space to another. The expression for interstitial impurity diffusion coefficient runs as follows: Dvi = D0viexp(-Q2i/kT)
respectively. As shown in Table 2, experimental values of volume selfdiffusion energy agree well with those calculated by formula (4). 2.2. Diffusion of interstitial atoms We shall apply the above approach to describe volume interstitial diffusion. As in case of volume self-diffusion, a liquid 'diffusion corridor' is required with atomic radius ri and length equal to the distance between neighboring interstitial sites. (This distance is generally similar to interatomic distance b). Interstitial diffusion activation energy Qvil can be written as follows:
(3)
where Dvi0 is the pre-exponential factor, Q2i is the interstitial atom migration energy. Rigorous calculations for vacancy formation energies Q1s and migration energies Q2s and Q2i are rather complicated based on the initial principles and generally require adjustable ill-defined parameters [14]. Therefore there is a whole series of phenomenological models proposed to assess diffusion activation energy values. However, despite much effort current methods are still not efficient enough to calculate volume diffusion parameters in pure metals [14]. Paper [15] proposes a way to describe grain boundary self-diffusion at high temperatures in melting terms (see earlier papers [16,17]). Paper [15] shows that in order to ensure diffusion mass transfer in the grain boundary it is enough to melt a small portion (in size close to an inter-atomic distance) of the grain boundary through fluctuations and to transfer the atom in the melt. This model helped to achieve reasonable values of grain boundary selfdiffusion activation energy for a wide range of pure metals. The core idea of this paper is to apply the approach developed in paper [15] to describe volume self-diffusion and interstitial diffusion and based thereon to calculate volume self-diffusion activation energy in FCC and BCC pure metals and interstitial impurity diffusion energy in BCC metals.
Qvil = Q mi + QLi + We
As above (see (5)), liquid 'diffusion corridor' formation energy Qmi can be calculated by the following formula: Qmi = πri2bλρ +2π(ri2+ rib)γS/L
(
We = K ΔV V
Similar to the way it was done in Ref. [15], let us assume that to ensure diffusion transfer of an atom in the crystal over a distance of the crystal lattice parameter, it is essential first to form a vacancy in the neighboring lattice site, second to melt a 'diffusion corridor' between a diffusing atom and a vacancy, and third to ensure migration of a diffusing atom along the resulting melt site. Diffusion activation energy value in this case may be presented as follows:
(9)
e
(ΔV/V)e =(ΔV/V)g - N∗(βΔТ + ΔVm/V)
(10)
∗
At Т = 0.5Tm and N = 6, ri = 0.77 nm, rp = 0.66 nm, Vm = 0.05, βΔТ∼0.05 we get (ΔV/V)e∼0.01. Subject to such (ΔV/V)e, We calculated by formula (9) with due regard to (10) at KΩ/kTm∼80 [20] is very small ∼10−3 kTm. This means that during formation of a 'diffusion corridor' by melting, the contribution of elastic strain energy We to Qlvi may be neglected. In this case the interstitial diffusion activation energy shall be determined by two members: energy required to create a 'diffusion corridor' Qmi and interstitial migration activation energy in the melt QLi. Theoretical values of the diffusion activation energy calculated by formula (7) for interstitial atoms in various metal systems are presented in Tables 3 and 4. Tables show that they agree satisfactorily with experimental data on diffusion of impurity atoms at high temperature [1–4].
(4)
where Q1s is the vacancy formation energy; Qms is the 'diffusion corridor' melting energy; QLs is the melt diffusion activation energy. To determine Q1s and QLs values, table values provided in Refs. [14] and [15,18] respectively can be used. To assess Qms, following the approaches developed in Ref. [15], let us present the 'diffusion corridor' formation enthalpy as a combination of two members: bulk melting energy and surface energy: (5)
where λ is the specific heat of melting, ρ is the mass density, γS/L is the liquid-crystal surface energy [15], V∗ and S∗ stand for the volume and area of the corridor surface. Let us assume that the molten area is shaped like a cylinder with atomic radius rs and length corresponding to interatomic distance b. In this case, V∗ = πrs2b and S∗ = 2π(rs2+ rsb). Plugging (5) in (4) with due regard to the above relations for V∗ and S∗ we shall get: Qms = πrs2bλρ+2π(rs2+ rsb)γS/L
2
) /2
where K is the bulk modulus of elasticity [18]. In turn, (ΔV/V)e values are composed of three members: geometric contribution (ΔV/V)g, thermal expansion contribution (ΔV/V)T and contribution related to volume jump during melting: (ΔV/V)e =(ΔV/ V)g - (ΔV/V)T - (ΔV/V)M. The geometric contribution (ΔV/V)g is equal to the difference in volume between an octahedral (tetrahedral) interstice Vp and an interstitial atom in BCC lattice Vi at zero temperature (ΔV/V)e=(Vp–Vi)/ Vp = 1-(ri/rp)3. The thermal expansion contribution at first approximation can be presented as follows: (ΔV/V)T = N∗βΔТ, where β is the coefficient of volume expansion, N∗ is the number of atoms surrounding the interstitial atom. The contribution (ΔV/V)M related to volume jump during melting ΔVm/V (for metals it is several percent ΔVm/V∼4–6% [15]) in the liquid 'diffusion corridor' equals N∗ΔVm/V. In this case the expression for (ΔV/V)e runs as follows:
2.1. Self-diffusion
Qms = λρV∗+γS/LS∗
(8)
where QLi is the interstitial diffusion activation energy in the melt; We is the elastic strain energy associated with dimensional discrepancy between an interstitial atom with radius ri and an interstice with radius rp where the atom is placed. At first approximation We is as follows:
2. Model of diffusion at high temperatures (T > TD)
1 Qvs = Q1s + Q ms + QLs
(7)
2.3. Discussions of the liquid diffusion corridor model Let us discuss the conditions for applicability of the above model. In line with Lindemann model [26], the atom oscillation energy in the crystal atoms with mа mass W ∼ mах2ν2/2 shall reach some limit value W∗. As shown in Ref. [26], this is possible when the frequency of atom oscillations ν reaches its limit value – the Debye frequency νD, and the amplitude of oscillations x reaches some limit value хmax. Chances to reach maximum frequency and amplitude of oscillations
(6)
Values of all the members constituting formula (4) that are calculated by formula (4), as well as volume diffusion activation energy values for a number of pure metals are presented in Tables 1 and 2 274
Materials Chemistry and Physics 219 (2018) 273–277
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Table 1 Basic parameters used in calculations. Material
Parameter b × 10−10, m [19]
TD, K
Cu
2.56
Al
2.86
310 347 433 380 477 432 423 365 276 250 606 405
α-Fe
2.5
Mo
2.73
Nb
2.86
Cr
2.5
[21] [20] [20] [21] [20] [21] [20] [21] [20] [21] [20] [21]
KΩ/kTm [20]
rs × 10−10, m [20]
95.9 102.4 81.4
Тm, K [20]
λρb3/kTm [15]
λS/Lb2/kTm [15]
QL/kTm [15]
Qls/kTm [14]
γSb2/kTm [20]
rp × 10−10, mb
1.28
1356
1.64
0.9
3.6
8.6
6
–
1.43
933
1.96
3.8
8.25
7.2
–
3.8 4.9 4
9
5.75
0.66–0.62
10
4.1
0.728
a
1.26
1810
1.16
1.19
c
c
c
0.764 1.12
c
98
1.4
2895
1.99
68.6
1.45
2760
1.59
0.944
4
10
4.81
0.75
78.2
1.27
2133
1.2
1
4
10
2.95
0.66
a
Atomic radii were used for calculations. In accordance with [12], it was assumed that an interstitial atom diffuses over large octahedral interstices. The size of the octahedral emptiness is determined by formula rp ∼ krs (where k = 0.46–0.52 is a geometric coefficient (see e.g. Ref. [22])). c Calculations were made in accordance with the calculations proposed in Ref. [15]. b
3. Diffusion model at low temperatures T < TD
largely depend on temperature. At temperatures T below the Debye temperature TD the amplitude of atom oscillations is minimum х < < хmax, whereas thermal energy passed over to atoms is spent mainly on increasing oscillation frequency which reaches maximum value νD at T = TD. At temperatures above the Debye temperature (T > TD) when maximum oscillation frequency νD is reached, temperature is rising together with oscillation amplitude that reaches хmax at melting temperature Tm. The Lindemann model provides obvious conditions for applicability of the melting model: W=W∗ condition is achievable only at temperatures above the Debye temperature when both the oscillation frequency and then the oscillation amplitude reach their maximum. In case of low temperatures (T ≤ TD) when the Lindemann condition is hard to meet, melting fluctuation becomes unlikely and the above model of diffusion through a liquid 'diffusion corridor' is inapplicable.
3.1. Self-diffusion model To describe diffusion at low temperatures T < TD we shall use the model of the 'diffusion corridor' shaped as a hollow cylinder. Let us assume that the movement of an atom inside such a corridor occurs without activation. Let us further assume that geometrical parameters of a hollow cylinder fully coincide with the parameters of a similar liquid cylinder. Under these assumptions, the expression for volume diffusion activation energy runs as follows:
Qhvs = 2π(r 2s + rsb)γs + Qls
(11)
where γs is the free surface energy. Theoretical values of volume self-diffusion activation energy Qhvs are provided in Table 2. As shown in Table 2, Qhvs values obtained by formula (11) are almost twice as high as experimental values Qv obtained at high temperature (see Table 2). This means that volume diffusion at low temperatures T < TD is difficult though the mechanism of hollow
Table 2 exp Theoretical (Qth v ) and experimental (Q v ) volume self-diffusion activation energy in various metal systems. Material
Parameter Т < TD, K
Qth v /kTm (formula (11))
Т > TD, K
Qth v /kTm (formula (4))
Qexp v /kTm
Cu
300
36.9
992–1355 973–1101
17.7
Al
300
42.2
673–883 603–733
19.1
α-Fe
300
32.3
1200–1500 1023–1163
17.4
Mo
300
30
1873–2473 2428–2813
19.3
Nb
200
31.8
1151–2668 1151–2668 1973–2373 1973–2373
19.8
Cr
300
24.2
1473–1873 1273–2023
19.8
18.8 [1,3,4] 16.2 [1] 21.6 [2] 16.9 [4] 18.6 [2] 16.5 [1] 15.7–22.3 [4] 16.9 [2,4] 19.8 [17] 15.9–20.1 [4] 17.5 [2]] 19.1 [1] 18.2–22.9 [4] 17.5 [1] 17.3 [3] 20.1 [1] 21.0 [1] 16.4–22.1 [4] 17.3 [2] 19.1 [1] 22.9–25.0 [4]
275
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E.S. Smirnova et al.
Table 3 exp Theoretical (Qth v ) and experimental (Q v ) activation energies of volume interstitial diffusion in various metal systems. Material
Parameter ri × 10−10, m [23]
a
( )
We kTm
Т < T D, K
Qth v /kTm formula (12)
Qexp v /kTm
Т > TD, K
Qth v /kTm formula (7)
Qexp v /kTm
470 470–480 508
7.5
8.2 [13] 7.2b 9.2 [13] 6.4b 5.4
ΔV V e
Fe-C
0.83
0.49
9.8
312.5–314
5.5
5.2–5.3 [13]
Fe-N
0.77
0.36
5.4
5
Mo-O
0.8
0.33
5.3
296–298 290 393
4.9 [13] 5.2 [13] 4.4 [24]
a b
4.4
473
7.1 6.3 7.2
Covalent radii of interstitial atoms were used for calculations. Average values of the diffusion activation energy in carbon and nitrogen atoms for all BCC metals specified in literature [1,2,13].
40
Table 4 exp Theoretical (Qth v ) and experimental (Q v ) volume interstitial diffusion activation energies in various metal systems at high temperatures Т > TD. Material
Parameter T, K
Qth v /kTm formula (7)
Qexp v /kTm
1473–1873 1100–2500 1100–2500 1473–1873 484–512 1118–1403 1203–2073
7.4 7.1
Nb-N
423–568 548–561
6.4
Cr-N
300–1573
6
7.1 5.7 8.3 6.6 5.8 6.3 6.9 6.2 6.4 6.4 7.1 6.0
Mo-C Mo-N Ta-C Ta-N V-C Nb-C
Q/kTm
6.6 6.3 6.4 6.7
30
[2] [1] [1] [1] [1] [1] [2] [3] [1] [1] [3] [25]
Qvs
20
Qvi
10
TD
0
cylinder formation.
0
0.2
T/Tm 0.4
0.6
Fig. 1. Temperature dependence of volume diffusion activation energy Qvs and interstitial impurity diffusion energy Qvi at low (Т < TD) and high (T > TD) temperatures. Scheme.
3.2. Interstitial diffusion Let us apply the procedure similar to the one set forth in the previous paragraph to describe low-temperature interstitial diffusion. Let us assume that like in case of volume self-diffusion, diffusion of interstitial atoms from one interstice into the neighboring one may be arranged by forming a hollow 'diffusion corridor'. The expression for Qhvi is then similar to (11). As mentioned above (see (9)), an interstitial atom is related to elastic strain energy We the release of which reduces the energy required for the atomic jump. At low temperatures We is much higher than at high temperatures. This is explained by at least two reasons. First, at low temperatures T < TD both the volume expansion coefficient β and the thermal expansion value are noticeably lower than at high temperatures (T > TD) [20]. Second, at T < TD contribution ΔVm/V associated with the lattice expansion during melting does not work because as the above description shows melting is difficult. Thus, (ΔV/V)e appears to approximate ∼(ΔV/V)g (see (10)), and the elastic contribution to the volume diffusion activation energy becomes rather significant (We reaches 5 kTm (see Table 3)). (Estimates for all members of (8) are also provided in Table 3.). In this case, the expression for the activation energy of volume diffusion of interstitial impurity atoms runs as follows:
Qhvi = 2πγs (r 2i + rib) − We
impossible. 4. Evaluation of results Fig. 1 schematically shows temperature dependences of volume diffusion activation energy values calculated by formulae (6), (7), (11), (12). As shown in Fig. 1, volume self-diffusion at low temperatures (T < TD) is rather difficult. At the same time interstitial diffusion at low temperatures (T < TD) is much easier than at high temperatures (T > TD). As shown above, self-diffusion activation energy at low temperatures depends only on the surface energy of the ′diffusion corridor'. Since the energy required to form a liquid-crystal interface during the formation of a liquid ′diffusion corridor' is twice as low as the free surface energy for a hollow ′diffusion corridor', volume self-diffusion activation energy in the liquid corridor model is lower in the hollow corridor model. However, at low temperatures T < TD when melting is difficult, volume self-diffusion activation energy appears to be rather high which means almost complete lack of self-diffusion at T < TD. Elastic strain energy plays a crucial role during interstitial diffusion. High temperatures cause significant thermal expansion of the lattice, therefore its contribution is small. However, at low temperatures the role of elastic energy grows substantially as during the formation of the hollow ′diffusion corridor' it promotes a substantial decrease in volume diffusion activation energy. It all leads to an important conclusion that interstitial diffusion at T < TD is possible but its activation energy is lower than at high temperatures T > TD.
(12)
Qhvi
calculated The values of interstitial diffusion activation energy by formula (12) are provided in Table 3. Let us emphasize that Qhvi values calculated by formula (12) are noticeably lower than corresponding values Qlvi obtained by formula (7) in the liquid ′diffusion corridor' model. It means that unlike volume self-diffusion, interstitial diffusion may occur at low temperatures as well when melting is almost 276
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4.1. Comparison with the experiment
1989, p. 510 (in Russian). [2] S. Mrowec, Defects and Diffusion in Solids. In Introduction, Elsevier, AmsterdamOxford-New York, 1980, p. 466. [3] H. Mehrer, Diffusion in Solids, Springer-Verlag, Berlin Heidelberg, 2007, p. 645. [4] G. Neumann, C. Tuijn, Self-diffusion and Impurity Diffusion in Pure Metals: Handbook of Experimental Data, Pergamon, 2009, p. 349. [5] Y. Zeng, Q. Li, K. Bai, Prediction of interstitial diffusion activation energies of nitrogen, oxygen, boron and carbon in bcc, fcc, and hcp metals using machine learning, Comput. Mater. Sci. 144 (2018) 232–247 https://doi.org/10.1016/j. commatsci.2017.12.030. [6] R. Mohammadzadeh, M. Mohammadzadeh, Texture dependence of hydrogen diffusion in nanocrystalline nickel by atomistic simulations, Int. J. Hydrogen Energy 43 (2018) 7117–7127 https://doi.org/10.1016/j.ijhydene.2018.02.145. [7] S.S. Naghavi, V.I. Hegde, C. Wolverton, Diffusion coefficients of transition metals in fcc cobalt, Acta Mater. 132 (2017) 467–478 https://doi.org/10.1016/j.actamat. 2017.04.060. [8] S. Divinski, Chapter 2 – Tracer Diffusion and Understanding the Atomic Mechanisms of Diffusion. - Handbook of Solid State Diffusion, Volume 1, Diffusion Fundamentals and Techniques, (2017), pp. 55–78. [9] R.A. Pérez, J.A. Gordillo, N. Di Lalla, U diffusion in IV-B elements in their HCPPhase, Procedia Materials Science 8 (2015) 861–867 https://doi.org/10.1016/ j.mspro.2015.04.146. [10] M. Christensena, W. Wolf, C. Freeman, et al., Diffusion of point defects, nucleation of dislocation loops, and effect of hydrogen in hcp-Zr: b initio and classical simulations, J. Nucl. Mater. 460 (2015) 82–96 https://doi.org/10.1016/j.jnucmat.2015. 02.013. [11] A.J. Ross, H.Z. Fang, S.L. Shang, et al., A curved pathway for oxygen interstitial diffusion in aluminum, Comput. Mater. Sci. 140 (2017) 47–54 https://doi.org/10. 1016/j.commatsci.2017.08.014. [12] A. Novik, B. Beri, Relaxation Phenomena in Crystals, in: E.M. Nagorny, YaM. Soyfer (Eds.), Atomizdat, Moscow, 1975, p. 472 (in Russian). [13] M.S. Blanter, YuV. Piguzov, G.M. Ashmarinet, et al., Internal Friction Method in Materials Science Studies, Metallurgiya, Moscow, 1991, p. 248 (in Russian). [14] A.N. Orlov, YuV. Trushin, Point defect Energy in Metals, Energoatomizdat, Moscow, 1983, p. 80 (in Russian). [15] V.N. Chuvil’deev, Non-equilibrium Grain Boundaries. Theory and Applications, Fizmatlit, Moscow, 2004, p. 304 (in Russian). [16] L.M. Klinger, Diffusion and heterophase fluctuations, Metallofizika 6 (5) (1984) 11–18 (in Russian). [17] K.A. Osipov, Some Activated Processes in Solid Metals and Alloys, Academy of Sciences of the USSR, Moscow, 1962, p. 131 (in Russian). [18] N.A. Vatolin (Ed.), Transport Properties of Metal and Slag Melts, Metallurgiya, Moscow, 1995, p. 649 (in Russian). [19] H.J. Frost, M.F. Ashby, Deformation-mechanism Maps: the Plasticity and Creep of Metals and Ceramics, Pergamon Press, New York, 1982, p. 166. [20] I.S. Grigoryev, E.Z. Meylikhov (Eds.), Physical Values. Reference Guide, Energoatomizdat, Мoscow, 1991, p. 1232 (in Russian). [21] E.M. Sokolovskaya, L.S. Guzey, Metallokhimiya, Moscow State University, Moscow, 1986, p. 264 (in Russian). [22] E.V. Chuprunov, A.F. Khokhlov, M.A. Faddeyev, Fundamentals of Crystallography. College Textbook, Fizmatlit, Moscow, 2004, p. 500 (in Russian). [23] S.S. Batsanov, Structural Chemistry. Facts and Figures, Dialogue-Moscow State University, Moscow, 2000, p. 292 (in Russian). [24] V.I. Baranova, S.A. Golovin, M.A. Krishtal, et al., Inelastic phenomena in molybdenum due to diffusion, Physics and Chemistry of Material Processing 2 (1968) 61–74 (in Russian). [25] B.A. Kolachev, YuV. Levinsky (Eds.), Coupling Constants of Metals and Gases. Reference Guide, Metallurgiya, Moscow, 1987, p. 368 (in Russian). [26] A.R. Ubbelohde, The Molten State of Matter: Melting and Crystal Structure, Willey, New York, 1978, p. 376. [27] A.A. Vasilyev, P.L. Gruzin, YuD. Zharov, E.S. Machurin, Recovery of internal friction of molybdenium single crystals, Phys. Met. Metallogr. 23 (2) (1967) 319–323 (in Russian). [28] R. Chalmers, W.P. Mason (Ed.), In “Physical Acoustics”, vol. 3A, Academic Press Inc, New York, 1966, p. 187.
Table 2 provides theoretical and experimental values for volume self-diffusion activation energies, Tables 3 and 4 give data on volume diffusion of interstitial impurities at low and high temperatures. As seen from Tables 3 and 4, most experimental data for volume interstitial diffusion activation energy Qvi was obtained at high temperatures (T > TD). Reliable data on Qvi at low temperatures (T < TD) are available only for three metal systems Fe-C, Fe-N and Mo-O. Amid lack of experimental data at low temperatures, another proof can be provided to support the above model – a link between Qvi and temperature experimentally observed in a number of papers [27,28]. Thus, papers [27,28] found that activation energy is lower at low temperatures rather than at high temperatures. This in essence corresponds to the results obtained here. (However, abnormally low values Qvi ≤ 1 kTm for low temperatures obtained in the said papers fail to comply with the typical values of interstitial diffusion activation energy in metals and therefore cannot be used in numerical calculations.) 5. Conclusion 1. This paper proposes two models for volume diffusion in metals operating at temperatures below the Debye temperature ('hollow diffusion corridor' model) and at temperatures above the Debye temperature ('liquid diffusion corridor' model). 2. Pursuant to the 'liquid diffusion corridor' model, atom migration at high temperatures is presented as diffusion motion of an atom in a molten cylindrical area towards the nearest vacancy. Theoretical values that were obtained for the activation energy of the volume self-diffusion in pure metals lie within the range of 18–20 kTm, while the values obtained for the activation energy of the interstitial impurity diffusion fall within the range of 6–7 kTm. 3. The 'hollow diffusion corridor' model views the migration process at temperatures below the Debye temperature when melting is unlikely as an activation less motion of an atom inside a hollow cylindrical area towards a nearby vacancy. This model shows that for interstitial impurities, the energy required to form a hollow diffusion corridor decreases due to the energy of elastic interaction between the interstitial atom and the interstice. The model helps to explain the effect observed during experiments – low values of the activation energy of the volume interstitial diffusion in metals at temperatures below the Debye temperature. In this case, theoretical values of the volume interstitial diffusion activation energy are 4–5 kTm. Acknowledgments This work was supported by the Russian Science Foundation (Grant No. 16-13-00066) and the Ministry of Education and Science of the Russian Federation (Grant No. 11.5944.2017). References [1] L.N. Larikov, V.I. Isaichev, Diffusion in Metals and Alloys, Naukova Dumka, Kiev,
277
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Mechanisms of volume diffusion in metals near the Debye temperature E.S. Smirnova, V.N. Chuvil'deev, A.V. Nokhrin
∗
T
Lobachevsky State University of Nizhny Novgorod, 603950, Nizhny Novgorod, Russia
H I GH L IG H T S
model is proposed at temperatures above (T > T ) and below (T < T ) the Debye temperature. • AAtdiffusion can be described in localized melting terms. • At TT >< TT diffusion occurs as a result of fluctuation formation of a hollow diffusion corridor. • The effect ofdiffusion reduced diffusion activation energy at low temperatures explained. • D
D
D D
A R T I C LE I N FO
A B S T R A C T
Keywords: Diffusion Metals and alloys Crystal structure Debye temperature
The paper offers a phenomenological model of volume self-diffusion and interstitial diffusion at high (T > TD) and low (T < TD) temperatures (where TD stands for the Debye temperature). Diffusion mechanisms at high and low temperatures were shown to differ greatly. Diffusion at high temperatures occurs as a result of fluctuations that can be described in the localized melting terms – 'liquid diffusion corridor' formation. At low temperatures when melting is difficult for a number of reasons, diffusion occurs through a 'hollow diffusion corridor' formed by fluctuation. Activation energy calculations for self-diffusion agree well with the T > TD experiment and show a dramatic increase in the activation energy at T < TD. Interstitial diffusion activation energy calculated for BCC metals agrees well with the experiment of the whole temperature range and helps to explain why diffusion activation energy goes down at low temperatures.
1. Introduction To date, experimental data on volume self-diffusion in various metal systems at high temperatures T ≥ 0.5Tm (Tm stands for material melting temperature) is extensive [1–11]. Experimental values of volume selfdiffusion activation energy Qvs that are generally obtained while studying diffusion permeability at 0.4–0.9Tm lie within the 18–20 kTm range [1–4]. A vacancy mechanism is traditionally used to provide a theoretical description of volume self-diffusion and interstitial diffusion. According to the classical concept of diffusion mass transfer mechanisms observed in the crystal lattice, a diffusing atom shall jump into the vacancy formed. This concept brings the following volume self-diffusion coefficient expression [2–4]: Dvs = Dvs0 exp(-(Q1s + Q2s)/kT)
(1)
Dvs0 = fza ν0exp((S1s + S2s))/k
(2)
2
where Dvs0 is the pre-exponential factor; f ∼ 1 is the correlation factor; ∗
z is the coordination number; ν0 is the frequency of atomic vibrations; Q1s and S1s are the energy and entropy of vacancy formation; Q2s and S2s are the energy and entropy of vacancy migration. Unlike volume self-diffusion, interstitial diffusion in BCC metals is experimentally studied over a wider temperature range: apart from Т≥0.4–0.5Tm temperatures generally used to study diffusion, interstitial diffusion was studied at lower temperatures (∼0.2Tm). As a rule, indirect methods are applied to this end. One of those methods is based on studying principles of internal friction [12,13]. It is traditionally assumed that interstitial diffusion is the reason for two internal friction peaks in BCC metals: Snoek peak at temperatures close to 0.2Tm and Snoek-Koster peak at 0.3–0.4Tm temperatures [12,13]. Standard methods help to identify interstitial diffusion activation energy and in some cases, the concentration of impurity atoms in a solid solution [1–4]. However, it should be emphasized that the values of diffusion activation energies in atoms of one and the same interstitial impurity obtained for those temperature ranges differ greatly [13]. Thus, when studying Snoek relaxation for carbon atoms in iron [13], the values of interstitial diffusion activation energy were obtained within the range
Corresponding author. E-mail address: [email protected] (A.V. Nokhrin).
https://doi.org/10.1016/j.matchemphys.2018.08.047 Received 11 November 2017; Received in revised form 9 June 2018; Accepted 19 August 2018 Available online 20 August 2018 0254-0584/ © 2018 Elsevier B.V. All rights reserved.
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of Qvi∼5.1–5.5 kTm, where as the Snoek-Koster relaxation values stood higher at Qvi∼8–9 kTm. An interstitial mechanism is used to provide a theoretical description for interstitial diffusion [2–4]. It is assumed that interstitial atoms need no vacancies for a diffusion jump as they spend energy only on migration from one interstitial space to another. The expression for interstitial impurity diffusion coefficient runs as follows: Dvi = D0viexp(-Q2i/kT)
respectively. As shown in Table 2, experimental values of volume selfdiffusion energy agree well with those calculated by formula (4). 2.2. Diffusion of interstitial atoms We shall apply the above approach to describe volume interstitial diffusion. As in case of volume self-diffusion, a liquid 'diffusion corridor' is required with atomic radius ri and length equal to the distance between neighboring interstitial sites. (This distance is generally similar to interatomic distance b). Interstitial diffusion activation energy Qvil can be written as follows:
(3)
where Dvi0 is the pre-exponential factor, Q2i is the interstitial atom migration energy. Rigorous calculations for vacancy formation energies Q1s and migration energies Q2s and Q2i are rather complicated based on the initial principles and generally require adjustable ill-defined parameters [14]. Therefore there is a whole series of phenomenological models proposed to assess diffusion activation energy values. However, despite much effort current methods are still not efficient enough to calculate volume diffusion parameters in pure metals [14]. Paper [15] proposes a way to describe grain boundary self-diffusion at high temperatures in melting terms (see earlier papers [16,17]). Paper [15] shows that in order to ensure diffusion mass transfer in the grain boundary it is enough to melt a small portion (in size close to an inter-atomic distance) of the grain boundary through fluctuations and to transfer the atom in the melt. This model helped to achieve reasonable values of grain boundary selfdiffusion activation energy for a wide range of pure metals. The core idea of this paper is to apply the approach developed in paper [15] to describe volume self-diffusion and interstitial diffusion and based thereon to calculate volume self-diffusion activation energy in FCC and BCC pure metals and interstitial impurity diffusion energy in BCC metals.
Qvil = Q mi + QLi + We
As above (see (5)), liquid 'diffusion corridor' formation energy Qmi can be calculated by the following formula: Qmi = πri2bλρ +2π(ri2+ rib)γS/L
(
We = K ΔV V
Similar to the way it was done in Ref. [15], let us assume that to ensure diffusion transfer of an atom in the crystal over a distance of the crystal lattice parameter, it is essential first to form a vacancy in the neighboring lattice site, second to melt a 'diffusion corridor' between a diffusing atom and a vacancy, and third to ensure migration of a diffusing atom along the resulting melt site. Diffusion activation energy value in this case may be presented as follows:
(9)
e
(ΔV/V)e =(ΔV/V)g - N∗(βΔТ + ΔVm/V)
(10)
∗
At Т = 0.5Tm and N = 6, ri = 0.77 nm, rp = 0.66 nm, Vm = 0.05, βΔТ∼0.05 we get (ΔV/V)e∼0.01. Subject to such (ΔV/V)e, We calculated by formula (9) with due regard to (10) at KΩ/kTm∼80 [20] is very small ∼10−3 kTm. This means that during formation of a 'diffusion corridor' by melting, the contribution of elastic strain energy We to Qlvi may be neglected. In this case the interstitial diffusion activation energy shall be determined by two members: energy required to create a 'diffusion corridor' Qmi and interstitial migration activation energy in the melt QLi. Theoretical values of the diffusion activation energy calculated by formula (7) for interstitial atoms in various metal systems are presented in Tables 3 and 4. Tables show that they agree satisfactorily with experimental data on diffusion of impurity atoms at high temperature [1–4].
(4)
where Q1s is the vacancy formation energy; Qms is the 'diffusion corridor' melting energy; QLs is the melt diffusion activation energy. To determine Q1s and QLs values, table values provided in Refs. [14] and [15,18] respectively can be used. To assess Qms, following the approaches developed in Ref. [15], let us present the 'diffusion corridor' formation enthalpy as a combination of two members: bulk melting energy and surface energy: (5)
where λ is the specific heat of melting, ρ is the mass density, γS/L is the liquid-crystal surface energy [15], V∗ and S∗ stand for the volume and area of the corridor surface. Let us assume that the molten area is shaped like a cylinder with atomic radius rs and length corresponding to interatomic distance b. In this case, V∗ = πrs2b and S∗ = 2π(rs2+ rsb). Plugging (5) in (4) with due regard to the above relations for V∗ and S∗ we shall get: Qms = πrs2bλρ+2π(rs2+ rsb)γS/L
2
) /2
where K is the bulk modulus of elasticity [18]. In turn, (ΔV/V)e values are composed of three members: geometric contribution (ΔV/V)g, thermal expansion contribution (ΔV/V)T and contribution related to volume jump during melting: (ΔV/V)e =(ΔV/ V)g - (ΔV/V)T - (ΔV/V)M. The geometric contribution (ΔV/V)g is equal to the difference in volume between an octahedral (tetrahedral) interstice Vp and an interstitial atom in BCC lattice Vi at zero temperature (ΔV/V)e=(Vp–Vi)/ Vp = 1-(ri/rp)3. The thermal expansion contribution at first approximation can be presented as follows: (ΔV/V)T = N∗βΔТ, where β is the coefficient of volume expansion, N∗ is the number of atoms surrounding the interstitial atom. The contribution (ΔV/V)M related to volume jump during melting ΔVm/V (for metals it is several percent ΔVm/V∼4–6% [15]) in the liquid 'diffusion corridor' equals N∗ΔVm/V. In this case the expression for (ΔV/V)e runs as follows:
2.1. Self-diffusion
Qms = λρV∗+γS/LS∗
(8)
where QLi is the interstitial diffusion activation energy in the melt; We is the elastic strain energy associated with dimensional discrepancy between an interstitial atom with radius ri and an interstice with radius rp where the atom is placed. At first approximation We is as follows:
2. Model of diffusion at high temperatures (T > TD)
1 Qvs = Q1s + Q ms + QLs
(7)
2.3. Discussions of the liquid diffusion corridor model Let us discuss the conditions for applicability of the above model. In line with Lindemann model [26], the atom oscillation energy in the crystal atoms with mа mass W ∼ mах2ν2/2 shall reach some limit value W∗. As shown in Ref. [26], this is possible when the frequency of atom oscillations ν reaches its limit value – the Debye frequency νD, and the amplitude of oscillations x reaches some limit value хmax. Chances to reach maximum frequency and amplitude of oscillations
(6)
Values of all the members constituting formula (4) that are calculated by formula (4), as well as volume diffusion activation energy values for a number of pure metals are presented in Tables 1 and 2 274
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Table 1 Basic parameters used in calculations. Material
Parameter b × 10−10, m [19]
TD, K
Cu
2.56
Al
2.86
310 347 433 380 477 432 423 365 276 250 606 405
α-Fe
2.5
Mo
2.73
Nb
2.86
Cr
2.5
[21] [20] [20] [21] [20] [21] [20] [21] [20] [21] [20] [21]
KΩ/kTm [20]
rs × 10−10, m [20]
95.9 102.4 81.4
Тm, K [20]
λρb3/kTm [15]
λS/Lb2/kTm [15]
QL/kTm [15]
Qls/kTm [14]
γSb2/kTm [20]
rp × 10−10, mb
1.28
1356
1.64
0.9
3.6
8.6
6
–
1.43
933
1.96
3.8
8.25
7.2
–
3.8 4.9 4
9
5.75
0.66–0.62
10
4.1
0.728
a
1.26
1810
1.16
1.19
c
c
c
0.764 1.12
c
98
1.4
2895
1.99
68.6
1.45
2760
1.59
0.944
4
10
4.81
0.75
78.2
1.27
2133
1.2
1
4
10
2.95
0.66
a
Atomic radii were used for calculations. In accordance with [12], it was assumed that an interstitial atom diffuses over large octahedral interstices. The size of the octahedral emptiness is determined by formula rp ∼ krs (where k = 0.46–0.52 is a geometric coefficient (see e.g. Ref. [22])). c Calculations were made in accordance with the calculations proposed in Ref. [15]. b
3. Diffusion model at low temperatures T < TD
largely depend on temperature. At temperatures T below the Debye temperature TD the amplitude of atom oscillations is minimum х < < хmax, whereas thermal energy passed over to atoms is spent mainly on increasing oscillation frequency which reaches maximum value νD at T = TD. At temperatures above the Debye temperature (T > TD) when maximum oscillation frequency νD is reached, temperature is rising together with oscillation amplitude that reaches хmax at melting temperature Tm. The Lindemann model provides obvious conditions for applicability of the melting model: W=W∗ condition is achievable only at temperatures above the Debye temperature when both the oscillation frequency and then the oscillation amplitude reach their maximum. In case of low temperatures (T ≤ TD) when the Lindemann condition is hard to meet, melting fluctuation becomes unlikely and the above model of diffusion through a liquid 'diffusion corridor' is inapplicable.
3.1. Self-diffusion model To describe diffusion at low temperatures T < TD we shall use the model of the 'diffusion corridor' shaped as a hollow cylinder. Let us assume that the movement of an atom inside such a corridor occurs without activation. Let us further assume that geometrical parameters of a hollow cylinder fully coincide with the parameters of a similar liquid cylinder. Under these assumptions, the expression for volume diffusion activation energy runs as follows:
Qhvs = 2π(r 2s + rsb)γs + Qls
(11)
where γs is the free surface energy. Theoretical values of volume self-diffusion activation energy Qhvs are provided in Table 2. As shown in Table 2, Qhvs values obtained by formula (11) are almost twice as high as experimental values Qv obtained at high temperature (see Table 2). This means that volume diffusion at low temperatures T < TD is difficult though the mechanism of hollow
Table 2 exp Theoretical (Qth v ) and experimental (Q v ) volume self-diffusion activation energy in various metal systems. Material
Parameter Т < TD, K
Qth v /kTm (formula (11))
Т > TD, K
Qth v /kTm (formula (4))
Qexp v /kTm
Cu
300
36.9
992–1355 973–1101
17.7
Al
300
42.2
673–883 603–733
19.1
α-Fe
300
32.3
1200–1500 1023–1163
17.4
Mo
300
30
1873–2473 2428–2813
19.3
Nb
200
31.8
1151–2668 1151–2668 1973–2373 1973–2373
19.8
Cr
300
24.2
1473–1873 1273–2023
19.8
18.8 [1,3,4] 16.2 [1] 21.6 [2] 16.9 [4] 18.6 [2] 16.5 [1] 15.7–22.3 [4] 16.9 [2,4] 19.8 [17] 15.9–20.1 [4] 17.5 [2]] 19.1 [1] 18.2–22.9 [4] 17.5 [1] 17.3 [3] 20.1 [1] 21.0 [1] 16.4–22.1 [4] 17.3 [2] 19.1 [1] 22.9–25.0 [4]
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Table 3 exp Theoretical (Qth v ) and experimental (Q v ) activation energies of volume interstitial diffusion in various metal systems. Material
Parameter ri × 10−10, m [23]
a
( )
We kTm
Т < T D, K
Qth v /kTm formula (12)
Qexp v /kTm
Т > TD, K
Qth v /kTm formula (7)
Qexp v /kTm
470 470–480 508
7.5
8.2 [13] 7.2b 9.2 [13] 6.4b 5.4
ΔV V e
Fe-C
0.83
0.49
9.8
312.5–314
5.5
5.2–5.3 [13]
Fe-N
0.77
0.36
5.4
5
Mo-O
0.8
0.33
5.3
296–298 290 393
4.9 [13] 5.2 [13] 4.4 [24]
a b
4.4
473
7.1 6.3 7.2
Covalent radii of interstitial atoms were used for calculations. Average values of the diffusion activation energy in carbon and nitrogen atoms for all BCC metals specified in literature [1,2,13].
40
Table 4 exp Theoretical (Qth v ) and experimental (Q v ) volume interstitial diffusion activation energies in various metal systems at high temperatures Т > TD. Material
Parameter T, K
Qth v /kTm formula (7)
Qexp v /kTm
1473–1873 1100–2500 1100–2500 1473–1873 484–512 1118–1403 1203–2073
7.4 7.1
Nb-N
423–568 548–561
6.4
Cr-N
300–1573
6
7.1 5.7 8.3 6.6 5.8 6.3 6.9 6.2 6.4 6.4 7.1 6.0
Mo-C Mo-N Ta-C Ta-N V-C Nb-C
Q/kTm
6.6 6.3 6.4 6.7
30
[2] [1] [1] [1] [1] [1] [2] [3] [1] [1] [3] [25]
Qvs
20
Qvi
10
TD
0
cylinder formation.
0
0.2
T/Tm 0.4
0.6
Fig. 1. Temperature dependence of volume diffusion activation energy Qvs and interstitial impurity diffusion energy Qvi at low (Т < TD) and high (T > TD) temperatures. Scheme.
3.2. Interstitial diffusion Let us apply the procedure similar to the one set forth in the previous paragraph to describe low-temperature interstitial diffusion. Let us assume that like in case of volume self-diffusion, diffusion of interstitial atoms from one interstice into the neighboring one may be arranged by forming a hollow 'diffusion corridor'. The expression for Qhvi is then similar to (11). As mentioned above (see (9)), an interstitial atom is related to elastic strain energy We the release of which reduces the energy required for the atomic jump. At low temperatures We is much higher than at high temperatures. This is explained by at least two reasons. First, at low temperatures T < TD both the volume expansion coefficient β and the thermal expansion value are noticeably lower than at high temperatures (T > TD) [20]. Second, at T < TD contribution ΔVm/V associated with the lattice expansion during melting does not work because as the above description shows melting is difficult. Thus, (ΔV/V)e appears to approximate ∼(ΔV/V)g (see (10)), and the elastic contribution to the volume diffusion activation energy becomes rather significant (We reaches 5 kTm (see Table 3)). (Estimates for all members of (8) are also provided in Table 3.). In this case, the expression for the activation energy of volume diffusion of interstitial impurity atoms runs as follows:
Qhvi = 2πγs (r 2i + rib) − We
impossible. 4. Evaluation of results Fig. 1 schematically shows temperature dependences of volume diffusion activation energy values calculated by formulae (6), (7), (11), (12). As shown in Fig. 1, volume self-diffusion at low temperatures (T < TD) is rather difficult. At the same time interstitial diffusion at low temperatures (T < TD) is much easier than at high temperatures (T > TD). As shown above, self-diffusion activation energy at low temperatures depends only on the surface energy of the ′diffusion corridor'. Since the energy required to form a liquid-crystal interface during the formation of a liquid ′diffusion corridor' is twice as low as the free surface energy for a hollow ′diffusion corridor', volume self-diffusion activation energy in the liquid corridor model is lower in the hollow corridor model. However, at low temperatures T < TD when melting is difficult, volume self-diffusion activation energy appears to be rather high which means almost complete lack of self-diffusion at T < TD. Elastic strain energy plays a crucial role during interstitial diffusion. High temperatures cause significant thermal expansion of the lattice, therefore its contribution is small. However, at low temperatures the role of elastic energy grows substantially as during the formation of the hollow ′diffusion corridor' it promotes a substantial decrease in volume diffusion activation energy. It all leads to an important conclusion that interstitial diffusion at T < TD is possible but its activation energy is lower than at high temperatures T > TD.
(12)
Qhvi
calculated The values of interstitial diffusion activation energy by formula (12) are provided in Table 3. Let us emphasize that Qhvi values calculated by formula (12) are noticeably lower than corresponding values Qlvi obtained by formula (7) in the liquid ′diffusion corridor' model. It means that unlike volume self-diffusion, interstitial diffusion may occur at low temperatures as well when melting is almost 276
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4.1. Comparison with the experiment
1989, p. 510 (in Russian). [2] S. Mrowec, Defects and Diffusion in Solids. In Introduction, Elsevier, AmsterdamOxford-New York, 1980, p. 466. [3] H. Mehrer, Diffusion in Solids, Springer-Verlag, Berlin Heidelberg, 2007, p. 645. [4] G. Neumann, C. Tuijn, Self-diffusion and Impurity Diffusion in Pure Metals: Handbook of Experimental Data, Pergamon, 2009, p. 349. [5] Y. Zeng, Q. Li, K. Bai, Prediction of interstitial diffusion activation energies of nitrogen, oxygen, boron and carbon in bcc, fcc, and hcp metals using machine learning, Comput. Mater. Sci. 144 (2018) 232–247 https://doi.org/10.1016/j. commatsci.2017.12.030. [6] R. Mohammadzadeh, M. Mohammadzadeh, Texture dependence of hydrogen diffusion in nanocrystalline nickel by atomistic simulations, Int. J. Hydrogen Energy 43 (2018) 7117–7127 https://doi.org/10.1016/j.ijhydene.2018.02.145. [7] S.S. Naghavi, V.I. Hegde, C. Wolverton, Diffusion coefficients of transition metals in fcc cobalt, Acta Mater. 132 (2017) 467–478 https://doi.org/10.1016/j.actamat. 2017.04.060. [8] S. Divinski, Chapter 2 – Tracer Diffusion and Understanding the Atomic Mechanisms of Diffusion. - Handbook of Solid State Diffusion, Volume 1, Diffusion Fundamentals and Techniques, (2017), pp. 55–78. [9] R.A. Pérez, J.A. Gordillo, N. Di Lalla, U diffusion in IV-B elements in their HCPPhase, Procedia Materials Science 8 (2015) 861–867 https://doi.org/10.1016/ j.mspro.2015.04.146. [10] M. Christensena, W. Wolf, C. Freeman, et al., Diffusion of point defects, nucleation of dislocation loops, and effect of hydrogen in hcp-Zr: b initio and classical simulations, J. Nucl. Mater. 460 (2015) 82–96 https://doi.org/10.1016/j.jnucmat.2015. 02.013. [11] A.J. Ross, H.Z. Fang, S.L. Shang, et al., A curved pathway for oxygen interstitial diffusion in aluminum, Comput. Mater. Sci. 140 (2017) 47–54 https://doi.org/10. 1016/j.commatsci.2017.08.014. [12] A. Novik, B. Beri, Relaxation Phenomena in Crystals, in: E.M. Nagorny, YaM. Soyfer (Eds.), Atomizdat, Moscow, 1975, p. 472 (in Russian). [13] M.S. Blanter, YuV. Piguzov, G.M. Ashmarinet, et al., Internal Friction Method in Materials Science Studies, Metallurgiya, Moscow, 1991, p. 248 (in Russian). [14] A.N. Orlov, YuV. Trushin, Point defect Energy in Metals, Energoatomizdat, Moscow, 1983, p. 80 (in Russian). [15] V.N. Chuvil’deev, Non-equilibrium Grain Boundaries. Theory and Applications, Fizmatlit, Moscow, 2004, p. 304 (in Russian). [16] L.M. Klinger, Diffusion and heterophase fluctuations, Metallofizika 6 (5) (1984) 11–18 (in Russian). [17] K.A. Osipov, Some Activated Processes in Solid Metals and Alloys, Academy of Sciences of the USSR, Moscow, 1962, p. 131 (in Russian). [18] N.A. Vatolin (Ed.), Transport Properties of Metal and Slag Melts, Metallurgiya, Moscow, 1995, p. 649 (in Russian). [19] H.J. Frost, M.F. Ashby, Deformation-mechanism Maps: the Plasticity and Creep of Metals and Ceramics, Pergamon Press, New York, 1982, p. 166. [20] I.S. Grigoryev, E.Z. Meylikhov (Eds.), Physical Values. Reference Guide, Energoatomizdat, Мoscow, 1991, p. 1232 (in Russian). [21] E.M. Sokolovskaya, L.S. Guzey, Metallokhimiya, Moscow State University, Moscow, 1986, p. 264 (in Russian). [22] E.V. Chuprunov, A.F. Khokhlov, M.A. Faddeyev, Fundamentals of Crystallography. College Textbook, Fizmatlit, Moscow, 2004, p. 500 (in Russian). [23] S.S. Batsanov, Structural Chemistry. Facts and Figures, Dialogue-Moscow State University, Moscow, 2000, p. 292 (in Russian). [24] V.I. Baranova, S.A. Golovin, M.A. Krishtal, et al., Inelastic phenomena in molybdenum due to diffusion, Physics and Chemistry of Material Processing 2 (1968) 61–74 (in Russian). [25] B.A. Kolachev, YuV. Levinsky (Eds.), Coupling Constants of Metals and Gases. Reference Guide, Metallurgiya, Moscow, 1987, p. 368 (in Russian). [26] A.R. Ubbelohde, The Molten State of Matter: Melting and Crystal Structure, Willey, New York, 1978, p. 376. [27] A.A. Vasilyev, P.L. Gruzin, YuD. Zharov, E.S. Machurin, Recovery of internal friction of molybdenium single crystals, Phys. Met. Metallogr. 23 (2) (1967) 319–323 (in Russian). [28] R. Chalmers, W.P. Mason (Ed.), In “Physical Acoustics”, vol. 3A, Academic Press Inc, New York, 1966, p. 187.
Table 2 provides theoretical and experimental values for volume self-diffusion activation energies, Tables 3 and 4 give data on volume diffusion of interstitial impurities at low and high temperatures. As seen from Tables 3 and 4, most experimental data for volume interstitial diffusion activation energy Qvi was obtained at high temperatures (T > TD). Reliable data on Qvi at low temperatures (T < TD) are available only for three metal systems Fe-C, Fe-N and Mo-O. Amid lack of experimental data at low temperatures, another proof can be provided to support the above model – a link between Qvi and temperature experimentally observed in a number of papers [27,28]. Thus, papers [27,28] found that activation energy is lower at low temperatures rather than at high temperatures. This in essence corresponds to the results obtained here. (However, abnormally low values Qvi ≤ 1 kTm for low temperatures obtained in the said papers fail to comply with the typical values of interstitial diffusion activation energy in metals and therefore cannot be used in numerical calculations.) 5. Conclusion 1. This paper proposes two models for volume diffusion in metals operating at temperatures below the Debye temperature ('hollow diffusion corridor' model) and at temperatures above the Debye temperature ('liquid diffusion corridor' model). 2. Pursuant to the 'liquid diffusion corridor' model, atom migration at high temperatures is presented as diffusion motion of an atom in a molten cylindrical area towards the nearest vacancy. Theoretical values that were obtained for the activation energy of the volume self-diffusion in pure metals lie within the range of 18–20 kTm, while the values obtained for the activation energy of the interstitial impurity diffusion fall within the range of 6–7 kTm. 3. The 'hollow diffusion corridor' model views the migration process at temperatures below the Debye temperature when melting is unlikely as an activation less motion of an atom inside a hollow cylindrical area towards a nearby vacancy. This model shows that for interstitial impurities, the energy required to form a hollow diffusion corridor decreases due to the energy of elastic interaction between the interstitial atom and the interstice. The model helps to explain the effect observed during experiments – low values of the activation energy of the volume interstitial diffusion in metals at temperatures below the Debye temperature. In this case, theoretical values of the volume interstitial diffusion activation energy are 4–5 kTm. Acknowledgments This work was supported by the Russian Science Foundation (Grant No. 16-13-00066) and the Ministry of Education and Science of the Russian Federation (Grant No. 11.5944.2017). References [1] L.N. Larikov, V.I. Isaichev, Diffusion in Metals and Alloys, Naukova Dumka, Kiev,
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