Effect of subgrain structure on the kinetics of phosphorus segregation in grain boundaries

Author’s Accepted Manuscript Effect of subgrain structure on the kinetics of phosphorus segregation in grain boundaries M.V. Sorokin, Z.V. Lavrukhina, A.N. Khodan, D.A. Maltsev, B.S. Bokstein, A.O. Rodin, A.I. Ryazanov, B.A. Gurovich www.elsevier.com

PII: DOI: Reference:

S0167-577X(15)30049-5 http://dx.doi.org/10.1016/j.matlet.2015.05.145 MLBLUE19039

To appear in: Materials Letters Received date: 20 March 2015 Revised date: 26 May 2015 Accepted date: 29 May 2015 Cite this article as: M.V. Sorokin, Z.V. Lavrukhina, A.N. Khodan, D.A. Maltsev, B.S. Bokstein, A.O. Rodin, A.I. Ryazanov and B.A. Gurovich, Effect of subgrain structure on the kinetics of phosphorus segregation in grain boundaries, Materials Letters, http://dx.doi.org/10.1016/j.matlet.2015.05.145 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Effect of subgrain structure on the kinetics of phosphorus segregation in grain boundaries.

M.V. Sorokin1,*, Z.V. Lavrukhina1, A.N. Khodan1, D.A. Maltsev1, B.S. Bokstein2, A.O. Rodin2, A.I. Ryazanov1, B.A. Gurovich1

1

2

National Research Centre 'Kurchatov Institute', Kurchatov Square 1, 123182 Moscow, Russia National University of Science and Technology "MISIS", Leninskiy prospekt 4, 119991

Moscow, Russia

Abstract The effect of steel microstructure, in particular the grain fragmentation onto subgrains, on the kinetics of phosphorus segregation in grain boundaries is considered. Assuming that intergranular phosphorus redistribution happens much faster than bulk diffusion, the segregation process is controlled by phosphorus transport toward the subgrain surfaces. The diffusion problem for a subgrain, allowing the phosphorus depletion in the subgrain volume, is proposed to describe the segregation kinetics. This model is applied to the pressure vessel steel 15Kh2NMFA-A and weld metal Sv-10KhGNMAA of the nuclear power reactor VVER-1000. It was shown that this model provides better fitting of the experimental data than the conventional Langmuir-McLean equation.

Keywords: Grain boundaries; Segregation kinetics; Reactor steels

*

Corresponding author, tel.: +7 495 517 46 89 e-mail: [email protected]

1

Introduction Segregations of the trace elements in the grain boundaries, in particular those of phosphorus, were considered as an origin of such effects as the intergranular cohesion decrease and temper brittleness in structural steels and so were intensively studied since 1950th [1-10]. The quantitative description of this effect is of great importance due to the problem of the maximal safe operation period of construction elements for nuclear power reactors. Kinetic model of phosphorus segregation at relatively low temperatures ~250-350 °C, proposed in the paper, was applied for the low carbon low-alloy vessel steel 15Kh2NMFA-A and weld metal Sv10KhGNMAA (Table 1) used in VVER-1000 reactors [11-14]. At the temperatures below 600 °C the phosphorus solubility in ferro-alloys drastically decreases [15], whereas the chemical stability of phosphor-containing precipitates in the grain boundaries is growing, driving formation of the segregations [16-18]. However, the phosphorus diffusion mobility also decreases, so that the segregation processes can be frozen kinetically. It makes difficult direct experimental study of the phenomenon at the temperatures below 500 °C and forces to assume some extrapolation of the experimental data for estimations of the longterm effect. Such estimations crucially depend on the implied kinetic models that can give different results based on the same set of the experimental data. The conventional model was proposed by McLean [2], who considered the impurity diffusion from the grain bulk and the Langmuir adsorption isotherm as a condition at the grain boundary. At the small adsorbate concentration at an interface the Langmuir equation reduces to the linear Henry law, which is commonly used. Nevertheless, the model is usually referred as Langmuir-McLean model. It provides the following expression for the temporal dependence of the phosphorus concentration in the grain boundary:   2 Dt  4 Dt  Cb (t ) = Cb (0) + (Cb (∞ ) − Cb (0 )) 1 − exp 2 2  erfc  s δ   sδ 

   

(1)

2

where Cb (0 ) is the initial and Cb (∞ ) is the equilibrium phosphorus concentration in the grain boundary, D is the volume diffusion coefficient of phosphorus, s is the enrichment coefficient assumed to be a constant, and δ is the width of the grain boundary. There are known issues of the Langmuir-McLean model, employing for description of the phosphorus segregation in the grain boundaries of different steels. In particular, as it can be seen in the Fig. 1 for the experimental data obtained at 550 °C [19], the Langmuir-McLean curve noticeably diverges from the experimental points in the long-time domain. The time required for saturation of the grain boundaries according to experimental data is significantly less than that according to the model, although the parameters of the Eq. (1), including the diffusion coefficient, are extracted by regression analysis from the same data [11, 19]. For the temperatures below 450 °С the main mechanism of the phosphorus transport is not quite clear, and kinetics of segregation in the grain boundaries remains insufficiently studied. However the similar discrepancy was observed in our study [11] of VVER-1000 pressure vessels steel 15Kh2NMFA-A and weld metal Sv-10KhGNMAA held at 310-320 °С, see Table 2 and Fig. 2. The possible origin of such difference can be the assumption of semi-infinite geometry of the diffusion problem, implied in the Eq. (1) without taking into account the real steel microstructure. As it should be for such geometry the phosphorus concentration asymptotically goes to the saturation value of Cb (∞ ) with the residue slowly decreases as ~ 1

t . The

experimentally observed saturation takes place much sharper that can be explained by the phosphorus depletion in the grain bulk. However such decrease of the phosphorus concentration in the whole grain volume is limited by the low value of the bulk diffusion coefficient. At the temperature 310-320 °C the diffusion length of phosphorous is just a few micrometers even for the longest available observation times of ~105 hours, so that the semi-infinite approximation of the LangmuirMcLean model can be justified for a ~50-micrometer grain. More convincing could be depletion of the certain vicinity of the grain boundary, which would be connected to the boundary via

3

dislocation networks. Being a rapid diffusion channels, the dislocations could provide intensive phosphorus transport to the grain boundaries [22-25]. Though such mechanism can work at the initial stage, it would be difficult to assume the dislocation connectivity deep enough to provide assured growth of the phosphorus segregations. Another mechanism is related with the internal structure of grains. Excistence of the subgrains and enhanced diffusion along their boundaries looks more promising and will be assumed in the present work. Both 15Kh2NMFA-A and Sv-10KhGNMAA steels have tempered bainite structure. The typical structure of the analysed samples is characterized by the average grain size of about 100 micrometers, although in the preceding austenite state the grains from 50 to 250 µm can be observed. Preliminary study of structures made by light microscopy and scanning electron microscopy (SEM) shows that the steel grains consist of 8 ± 3 fragments (Fig. 3a), but highmagnification SEM (Fig. 3b) reveals that each of them in turn consists of ~105 even smaller structure units – the subgrains with the size of about 2 µm [26, 27]. The boundaries between the subgrains are the enhanced diffusion channels piercing the whole grain, so that the segregation process is controlled by the phosphorous diffusion toward the surfaces of subgrains. Assuming its fast redistribution within the intergranular space we should have equal chemical potentials of phosphorus in the subgrain and the grain boundaries. Therefore the correspondent activities can differ only by a constant factor, which does not effect on the kinetic curves and can be omitted. Likewise in the case of a dilute phosphorus solution the temporal dependence of the phosphorus concentration in subgrain boundaries to be found from the diffusion problem for an isolated subgrain can be considered as Cb (t ) , i.e. compared and fitted with the quantity of phosphorus measured by Auger electron spectroscopy (AES) techniques on the facet cleaved along a grain boundary [9] (Table 2). It should be noted that the surface concentration, measured by AES, is often expressed in fractions of a monolayer (ML) [14]. As a first approximation it can be considered as the atomic concentration Cb multiplied by the number of the atomic layers (crystal planes) within the 4

boundary of width δ . From the other hand, the material depth, contributing to the AES spectra reaches 0.4 - 1.0 nm for PLVV and 1.0 - 1.3 nm for FeLVV electrons [21], which is comparable with δ . Thus we can consider that the experimental values in the Table 2 correspond to Cb in the model.

Model equations In order to develop analytical description of the model we will consider a subgrain as an isotropic sphere of radius R . Let C (r , t ) be the atomic concentration of phosphorus in the subgrain. In the spherical geometry it will obey the following diffusion equation:

∂C 1 ∂  2 ∂C  = r D , 0≤r < R ∂t r 2 ∂r  ∂r 

(2)

where D is the diffusion coefficient of phosphorus in the bulk. Here we do not consider any effects of external or intrinsic elastic stresses [28]. Ingress of phosphorus into the intergranular space assumed to be an unimpeded process, controlled by the diffusional supply. Thus the prosphorus concentration in the boundary vicinity C r = R corresponds to the equilibrium with the concentration in the the boundary Cb (t ) ; and the Dirichlet boundary condition can be applied for the diffusion problem (2). Similarly to the Langmuir-McLean model here we consider the linear domain of the Langmuir isotherm (Henry law) [9]

Cb = s C r = R , t > 0

(3)

where s is the enrichment coefficient (Henry constant) at the retention temperature. Initial phosphorus concentrations in the grain bulk and within the intergranular space are determined by the quenched high-temperature phosphorus distribution. C = C0 , t = 0 , r ≤ R Cb = s0 C0 , t = 0

(4)

5

where s0 < s is the enrichment factor at the quenching temperature. Boundary condition (3) for the Eq. (2) corresponds to the phosphorus balance:

δ ∂Cb = 2 jR ωb ∂t

(5)

where δ is the width of the subgrain boundary or some effective dimension of the intergranular space. Note that although the Henry law (3) assumes the ‘chemical’ equilibrium between the boundary and its close vicinity, the latter is not diffusional equilibrium towards the bulk; and the phosphorus current at the boundary j R is given by the Fick’s law:

jR = −

D ∂C ω ∂r

(6) r =R

Here ω and ωb are the atomic volumes in the bulk and within the intergranular space correspondingly. A factor 2 in the Eq. (5) takes into account the bilateral currents from the neighbor grains. Thus, combining Eqs. (5), (6) and (3) one can obtain:

δ ω ∂C ∂C s =D , r=R, t>0 2 ωb ∂t ∂r

(7)

Solution of the Eq. (2) supplemented with conditions (4) and (7) can be found similarly to the thermal conductivity problem for a sphere immersed in an ideal heat conductor (e.g. a stirring liquid) [29].

 3δ s    0 + 1 2 2 ∞  2R δ (s − s0 ) exp − Dα n t R sin (α n r R )  C (r , t ) = C0  − ∑  2 2 2 3δ s r 3δ s  n =1  δ s α n   +1   ( ) + + 1 sin α n  2   2R 2R  4R   

(

)

(8)

where α n are the roots of the equation tan α = 1+

α δs 2R

α2

(9)

Introducing the ‘phosphorus capacities’ of the intergranular space at the working and annealing temperatures as 6

k=

3δ s0 3δ s and k 0 = 2R 2R

(10)

the Eq. (9) can be rewritten in the form tan α = and Eq

3α 3 + kα 2

(11)

(8) gives the temporal dependence of the desired intergranular phosphorus

concentration:

k +1 exp(− Dα n t R )  Cb (t ) = sC r = R = sC0  0 − 6(k − k 0 )∑ 2 2 n =1 k α n + 9(k + 1)   k +1 

∞

2

2



(12)

Discussion Both Eqs. (1) and (12) are obtained in the assumption of the Henry law (3), when the grain boundary phosphorus concentrations formally can grow without saturation. Langmuir adsorption isotherm [17] implies a certain value Cbmax , limiting the number of sites in the boundary, which can be occupied by phosphorus atoms. So adsorption and desorption rates can

(

)

be written as p+ Cbmax − Cb C r = R and p− Cb respectively, where p+ and p− are coefficients depending on temperature. Such model gives non-linear boundary condition at r = R , and Eq. (2) can not be solved analyticaly. However for the self-adjusting diffusion profile of phosphorus the balance equation can be simplified as

dCb = p+* Cbmax − Cb C − p− Cb dt

(

)

(13)

where the modified coefficient p+* takes into account diffusion so that p+* C = p+ C r = R . With the conservation law C = C0 −

Vb (Cb − Cb (0)) , where V and Vb are the volumes of the grain and its V

boundary respectively, Eq. (13) gives the temporal dependence of the grain boundary phosphorus concentration as an easy to obtain but quite cumbersome expression. In the case Vb V << 1 the limiting value at t → ∞ is determined by the Langmuir equation:

7

Cb (∞ ) =

C0Cbmax C0 + p− p+*

(14)

as it should be for the non-limited phosphorus reserve in the grain, but the saturation of the boundary. Further experiments with different initial phosphorus content are needed in order to distinguish the main reason of the experimental curve saturation. Equation (12) describes kinetics of the phosphorous accumulation, taking into account the finite size of the subgrains R which is much smaller than the grain dimensions. From the other point of view it can be considered in terms of the enhanced diffusion through the whole grain volume as soon as the ratio D R 2 in the Eq (12) remains the same. What is important, the proposed model describes the phosphorus depletion in the bulk (Fig. 4). Thus the maximal phosphorous concentration in the grain boundary is limited by the value of sC0 (k 0 + 1) (k + 1) , whereas the Langmuir-McLean approach gives Cb (∞ ) = sC0 . Such difference is especially important for high-temperature kinetics, where the Eq (12) describes the long-time domain better than the Eq. (1) (see Fig. 1). In the case of 310 - 320 °C, when the bulk diffusion coefficient can be estimated to be as small as 10-18 − 10-19 cm2/s, predictions of the both models can be distinguished only for the retention times far beyond the experimentally available ones (Fig. 2). Nevertheless the understanding of the effect remains important and can become apparent e.g. under irradiation.

Acknowledgement to Russian Foundation of Basic Research for financial support (project # 13-03-12191_ofi_m)

References 1. F.L. Carr, М. Goldman, L.D. Jaffe and D.С. Вuffum, Trans. AIME 197 (1953) 998. 2. D. McLean, Grain boundaries in metals (Clarendon Press, Oxford, 1957). 8

3. М. Р. Seah and Е. О. Hondros, Proc. R. Soc. London (А) 335 (1973) 191. 4. М. Guttmann, Surface Sci. 53 (1975) 213. 5. Е.D. Hondros, М.Р. Seah, Int. Меt. Rev. 22 (1977) 262. 6. S.G. Park, K.H. Lee, K.D. Min, M.C. Kim, B.S. Lee, J. Nucl. Mater. 426 (2012) 1. 7. P. Lejcek, Grain Boundary Segregation in Metals (Springer, Berlin Heidelberg, 2010) 8. W. Losch, Acta Metall. 27 (1979) 1885. 9. V.V.Slezov, L.N.Davydov, V.V.Rogozhkin, Fizika tverdogo tela 12 (1995) 3565 (in Russian). 10. G.S. Was, Fundamentals of Radiation Material Science. Metals and Alloys (Springer, Berlin Heidelberg, 2007) 11. B.S. Bokshtein, A.N. Khodan, O.O. Zabusov, D.A. Mal’tsev, B.A. Gurovich, Phys. Met. Metallogr. 115 (2014) 146. 12. Y.I. Shtrombakh, B.A. Gurovich, E.A. Kuleshova, D.A. Mal'tsev, S. V. Fedotova, A.A. Chernobaeva, J. Nucl. Mater. 452 (2014) 348. 13. Y.I. Shtrombakh, B.A. Gurovich, E.A. Kuleshova, A.S. Frolov, D.Yu. Erak, D.A. Zhurko, E.V. Krikun, Atomic Energy 116 (2014) 373. 14. B. Gurovich, E. Kuleshova, Ya. Shtrombakh, S. Fedotova, D. Erak, D. Zhurko, J. Nucl. Mater. 456 (2015) 373. 15. H. Okamoto, The Fe-P (Iron-Phosphorus) System. Bulletin of Alloy Phase Diagrams 11 (1990) 404 http://dx.doi.org/10.1007/BF02843320 16. R.W. Balluffi, S.M. Allen, W.C. Carter, Kinetics of Materials (Wiley, Hoboken, 2005) 17. K.A. Jackson, Kinetic Processes. Crystal Growth, Diffusion, and Phase Transitions in Materials. 2nd edition (Wiley-VCH, Weinheim, 2010)

18. G. Gottstein, Physical Foundations of Materials Science (Springer, Berlin, Heidelberg, 2004).

9

19. L.M. Utevskii, E.E. Glikman, and G.S. Kark, Reversible Temper Brittleness of Steel and Alloys of Iron (Metallurgiya, Moscow, 1987) [in Russian]

20. D. Briggs, M.P. Seah, Practical Surface Analysis: Auger and X-ray photoelectron spectroscopy (Wiley & Sons, New York, 1990)

21. M. Prutton, M.M. El Gomati, Scanning Auger Electron Microscopy (Wiley & Sons, Chichester, 2006) 22. A.D. Le Claire and A. Rabinovitch, J. Phys. C: Solid State Phys. 14 (1981) 3863 23. A.D. Le Claire and A. Rabinovitch, J. Phys. C: Solid State Phys. 15 (1982) 3455 24. A.D. Le Claire and A. Rabinovitch, J. Phys. C: Solid State Phys. 16 (1983) 2087 25. H. Mehrer, Diffusion in Solids. Fundamentals, Methods, Materials, Diffusion-Controlled Processes (Springer Berlin, Heidelberg, 2007).

26. S.S.G. Park, K.H. Lee, K.D. Min, M.C. Kim and B.S. Lee, Met. Mater. Int., 19 (2013) 49. 27. B.Z. Margolin, V.A. Nikolayev, E.V. Yurchenko, Yu.A. Nikolayev, D.Yu. Erak, A.V. Nikolayeva, International Journal of Pressure Vessels and Piping 89 (2012) 178. 28. M.V. Sorokin, A.I. Ryazanov, J. Nucl. Mater. 357/1-3 (2006) 82. 29. H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, 1959). 30. J. Crank, The Mathematics of Diffusion. 2nd edition (Oxford University Press, London, 1975). 31. A.G. Sveshnikov, A.N. Tikhonov, The theory of functions of a complex variable (Mir Publishers, Moscow, 1978)

10

Table 1. Average chemical composition of the base metal samples (15Kh2NMFA-A) and the weld metal samples (Sv-10KhGNMAA.) Chemical composition, wt. % C Ni P Cu S Mn Si Cr Mo V 15Kh2NMFA-A 0.17 1.18 0.010 0.05 0.012 0.39 0.26 2.02 0.52 0.08 Sv-10KhGNMAA 0.06 1.64 0.008 0.03 0.008 0.76 0.36 1.72 0.63 − Type of steel

Table 2. Averaged phosphorus concentration in grain boundaries Cb [ML] measured by AES in the base and weld metal samples retained at 310-320 °C. t,

Steel samples

hours

15Kh2NMFA-A Sv-10KhGNMAA

0 6.70×10

4

1.37×10

5

1.95×10

5

0.140±0.04

0.134±0.03

0.141±0.03

0.166±0.03

0.177±0.04

0.168±0.04

0.176±0.04

0.185±0.03

Figure captions

Fig. 1. Kinetics of grain boundary saturation by phosphorus at 550 °C. Experimental data are adopted from [19]. Fig. 2. Temporal dependence of phosphorus concentration in grain boundaries for 15Kh2NMFAA (a) and Sv-10KhGNMAA (b) samples at 310 - 320 °C (according to [11]). Fig. 3. Low (a) and high (b) magnification SEM images of the Sv-10KhGNMAA steel sample. Subgrains of ~ 2 µ are clearly visible at high magnification. Fig. 4. Phosphorus diffusion profiles in a subgrain, corresponding to the Fig. 2a.

11

1,0

Cb , ML

0,8

experimental data proposed model Langmuir-McLean

0,6

0,4

0,2 0

5

10

15

20

t , hours

Figure 1

12

a)

0.30

6

0.25

5

0.20

4

experiment data proposed model Lengmuir- McLean

0.15

Cb, at%

Cb, ML

0.35

3

2

0.10 6

0

1x10

6

2x10

6

3x10

6

4x10

t, hours

0.35

7

0.30

6

0.25

5

0.20

4

experimental data proposed model Langmuir-McLean

0.15

0.10

0

6

1x10

6

2x10

6

3x10

Cb, at%

Cb, ML

b)

3

4x10

6

t, hours

Figure 2

13

Figure 3

14

0.020

C , at. %

0.015

0.010

0.005

0.000 0.0

t , hours 4 3*10 5 8*10 6 3*10 7 1*10 7 3*10

0.5

1.0

1.5

2.0

r,µ

Figure 4

highlight We model phosphorus segregations taking into account the sub-grain structure. The volume diffusion of phosphorus is assumed to be a limiting process. Diffusion problem for a subgrain has to account the phosphorus depletion in the bulk.

15