Anisotropy of interstitial diffusion in bcc-crystals due to stress-induced unequal occupancies of different types of sites

Accepted Manuscript

Anisotropy of interstitial diffusion in bcc-crystals due to stress-induced unequal occupancies of different types of sites J. Svoboda , F.D. Fischer PII: DOI: Reference:

S0020-7683(18)30217-8 10.1016/j.ijsolstr.2018.05.023 SAS 10003

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

25 January 2018 15 May 2018 27 May 2018

Please cite this article as: J. Svoboda , F.D. Fischer , Anisotropy of interstitial diffusion in bcc-crystals due to stress-induced unequal occupancies of different types of sites, International Journal of Solids and Structures (2018), doi: 10.1016/j.ijsolstr.2018.05.023

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ACCEPTED MANUSCRIPT

Anisotropy of interstitial diffusion in bcc-crystals due to stress-induced unequal occupancies of different types of sites J. Svobodaa), F.D. Fischerb)*) a)

Institute of Physics of Materials, Academy of Science of the Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic b) Institute of Mechanics, Montanuniversität Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria

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Abstract

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Deposition of an interstitial atom in octahedral or tetrahedral sites in a bcc-crystal provokes one of three types of local tensorial eigenstrains. Interaction of the interstitial atoms with an external and/or defect-generated stress state and their diffusion cause different occupancies of individual types of sites. The diffusion paths are analyzed for atoms occupying octahedral or tetrahedral sites. The current original model quantifies the anisotropy of diffusion by factors being functions of occupancies of individual types of sites. Coupling of this new model with a very recent model of interstitial diffusion, already accounting for various types of interstitial sites, provides a rather sophisticated theoretical model for simulation of interstitial diffusion in stressed crystals.

Keywords: Micro-mechanics, Inclusions, Interaction, Microstructural, Interstitial diffusion

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1. Introduction

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The following study deals with diffusion of interstitial atoms as impurity interstitials. From the practical point of view such impurity interstitials as C- and H-atoms play the most important role in Fe-alloys with bcc-crystal lattice, undoubtedly the most important industrial material system. Consequently we demonstrate our new diffusion concept for such a material system. However, we would like to point already now to the fact that the concept is also applicable for other material systems. Both, experimental investigations and theoretical research, have brought out that H-atoms prefer tetrahedral interstitials sites (tis), see, e.g., [113], and C-atoms prefer octahedral interstitial sites (ois), see, e.g., [14,15] and according later computational results [16-18]. One can distinguish 3 types of ois or tis depending on the orientation of the tetrahedron or octahedron with respect to the bcc-lattice, see Fig. 1. The types of ios and tis are denoted by numbers. The interstitial atoms can be considered as small misfitting inclusions provoking a local tensorial eigenstrain state and an according eigenstress state. This eigenstrain state interacts with the stress state due to external load and/or defects as dislocations, cracks, and other misfitting atoms etc. Here we refer to the pioneering works by Zener [19] and Cochardt et al. [20] (for details see [16]) more than six decades ago and to Flynn [21], Sect. 9 of his book. If one considers the interstitial atom as an inclusion of the volume of one atom, the respective eigenstrains in the coordinate system, coinciding with main crystallographic directions, assume remarkable values, e.g. e11  0.76 , e22  e33  0.05 for C-atom (measured as engineering strain components) in ois of type 1, see the recent study [22]. Analogously the eigenstrains assume e22  0.76 , e11  e33  0.05 for C-atom in ois of type 2 and e33  0.76 , *)

corresponding author: [email protected] 1

ACCEPTED MANUSCRIPT e11  e22  0.05 in ois of type 3. The values e11  0.04 , e22  e33  0.09 can be estimated by ab-initio method [23] for H-atom in tis of type 1, e22  0.04 , e11  e33  0.09 in tis of type 2 and e33  0.04 , e11  e22  0.09 in tis of type 3. Japanese researchers have already shown in ab

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initio studies [10,11] that such an anisotropy in the eigenstrain state may have also a strong influence on the activation energy of diffusion.

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a) b) Fig. 1. The bcc-lattice unit cell with a) ois and b) tis and numbers denoting their type. The diffusion paths along nearest neighbours are visualized by red double arrows (for tis only two diffusion paths are shown corresponding to diffusion in 1-direction). The mechanical part of the generalized chemical potential  * of the interstitial atom is

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given by  11e11   22e22   33e33   (  ij are the stress components of the total stress tensor excluding the misfit eigenstress state of the considered interstitial atom and  the molar volume), see [24], Sect. IV. 12, the former works [25,26], and for details with respect to the derivation see [27]. The interstitial atoms are distributed to the individual types of interstitial sites due to the different mechanical terms in the chemical potential. Their occupancies can be described by site fractions X i for i  1, 2,3 representing the type of interstitial sites. The

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values of the site fractions X i can be determined by assuming local thermodynamic equilibrium presented in [27]. Consequently the diffusion and mass conservation laws are formulated in site fractions X i for the first time by multiplying the conventional equation for the diffusive flux by the factor



1

1



1 1

 X1 1  X1    X 2 1  X 2    X 3 1  X 3   X1  X 2  X 3  . Finding these relation was one of the goals of the previous paper [27]. This concept is applicable for both ois and tis, although it has been demonstrated on an example only for H in ois. Moreover, it is necessary to point out, that the diffusion theory in [27] is formulated phenomenologically, i.e. without accounting for the features of the lattice structure, and thus exhibiting no anisotropy. The diffusion theory in [27] can be significantly improved by accounting for the features of the lattice structure. Both the local eigenstrain state and the stress field lead to a deformation of the bcclattice and cause an anisotropy of diffusion coefficient itself. Here, we refer to the pioneering works [21,26] already four decades ago and the recent excellent overview by Trinkle [28] 2

ACCEPTED MANUSCRIPT dealing with elastodiffusion. Very recent research papers as [29] show that diffusion in stressed material is still of practical and theoretical relevance. Going back to Flynn [21] and Dederichs and Schroeder [26] the diffusion coefficient is described by a tensor as Dij  D  ij  Dijel  ,

(1)

with D being the scalar diffusion coefficient in an unstrained lattice,  ij the unity tensor and DDijel being the elastodiffusion tensor. According to Flynn [21], Sect. 9, the dimension-free

tensor Dijel can be formulated as the sum of   H  ij with  H  11   22   33  3 and a

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tensor  ij with     for the diagonal terms and    for the off-diagonal terms. The coefficients  RgT ,   RgT ,   RgT ( Rg is the gas constant and T the absolute temperature), representing the activation volumes, can be taken from [21], Sect. 9. Most recently Trinkle [28] offered detailed and improved derivations of Dijel using the product of a forth order tensor dijkl with the strain tensor  kl given by the sum of an eigenstrain tensor and

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elastic strain tensor. Detailed derivations of dijkl are provided by Trinkle [28] for different types of lattices inclusive the bcc-lattice. For negligible eigenstrains, the strain tensor  kl can

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be calculated from the stress tensor  ij via the compliance tensor Sijkl , and Dijel is formulated in activation volumes again. It is necessary to point out that the effect of the stress state on the diffusion coefficient and on its anisotropy (elastodiffusion) expresses its influence on jump frequencies and/or jump distances. This effect is reflected by Eq. (1) and we do not deal with that problem in this paper (we consider this problem as already solved). However, the intensity and anisotropy in the diffusion process is caused not only by elastodiffusion, but also by different occupancies of the individual types of ois or tis, expressed by different local values of site fractions X i , i  1, 2,3 , in ois or tis). The actual

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values of X i , i  1, 2,3 , can be considered as known quantities, as demonstrated in detail in the preceding paper [27]. However, to the best knowledge of the authors, models accounting for the paths of interstitials along the tis or ois, quantifying the effect of different occupancies of the sites on the diffusion process, are still missing. The goal of the current paper is to provide the community with a proper treatment of this type of anisotropy by means of a set of diffusion occupancy factors representing adaption terms for diffusion fluxes. 2. System description

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As first step let us study the diffusion paths for ois and tis by using the unit cell with lattice parameter a shown in Figs. 1a (for ois) and 1b (for tis). The diffusion paths are indicated by red double arrows. We assume that diffusion occurs exclusively by jumping between nearest neighbouring ois or tis as following: ois: Each ois has 4 nearest neighbours in the distance of a 2 . The neighbouring ois form chains in main crystallographic directions. The chains in 1-direction consist of ois of alternating types 2 and 3, the chains in 2-direction consist of alternating types 1 and 3 and the chains in 3-direction of alternating of types 1 and 2, see Fig. 1a. Altogether, there are three admissible ois diffusion paths along straight lines oriented in mean crystallographic directions. 3

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tis: Each tis has 4 nearest neighbours in the distance of 2a 4 , and there are 4 tis on each face of the unit cell. The path of diffusion in 1-direction is rather complicated consisting of jumping in four directions between all types of nearest-neighbouring tis as it is shown in Fig. 1b. 3. Model for interstitial diffusion

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In the linear theory of diffusion, the diffusive flux of any orientation can be obtained by superposition of all the contributions from individual diffusion paths. The equations for diffusive fluxes can be derived from the models outlined below: 3.1. The ois-case

For the diffusion path in 1-direction the neighbouring ois are of alternating types 2 and 3. The probability per time unit for jumping of the atom from ois of type 2 to nearest neighboring ois of type 3 is given by

X 2 1  X 3  .

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p1,23

(2)

Eq. (2) expresses the fact that a jump may occur only if the ois of type 2 is occupied by an atom (the probability is X 2 ) and the nearest neighboring ois of type 3 is free (the probability

p1,32

X 3 1  X 2  .

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is 1  X 3 ). Analogously, the probability per time unit for jumping of the atom from ois of type 3 to nearest neighboring ois of type 2 is given by (3)

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Atoms diffusing along 1-direction face the same number of ois of type 2 and type 3. The mean time for jumping from ois of type 2 to the nearest neighboring ois of type 3 is given by t1,23  1 p1,23 1  X 2 1  X 3  .

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(4)

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The mean time for jumping from ois of type 3 to the nearest neighboring ois of type 2 is given by t1,32  1 p1,32 1  X 3 1  X 2  .

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(5)

Thus, the mean time for two jumps along 1-direction follows as t1  t1,23  t1,32 ,

(6)

and the respective probability per time unit reads as 1

p1  1 t1

  1 1    .  X 2 1  X 3  X 3 1  X 2  

(7)

Since other diffusion paths are orthogonal to the 1-direction, they cannot contribute. 4

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 x1 as *

1

  1 1 * j1   Aois p1  x1   Aois     x1 . X 1  X X 1  X     3 3 2   2 *

(8)

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The pre-multiplier Aois involves quantities related to frequency and geometry of diffusion jumping. For the case of a stress free dilute ideal solution system the diffusive flux is given by Fick's first law as

j1   D c x1 ,

(9)

which can be utilized for the calibration of the model (determination of Aois ). In the stress

j1   X

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free dilute system 3 X1  3 X 2  3 X 3  X  c  1 and  *  0  RgT ln X , and Eq. (9) can be rewritten as

D  * . RgT  x1

(10)

Comparison of Eqs. (8) and (10) allows evaluating Aois  6D  RgT   . Consequently, one can

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rewrite Eq. (8) as

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  1 1 D j1  6    * x1 .   X 2 1  X 3  X 3 1  X 2   RgT 

(11)

  1 1  6    X 2 1  X 3  X 3 1  X 2  

1

(12.1)

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F1,ois

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We denominate the factor

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as diffusion occupancy factor related to the flux in 1-direction for ois. Analogously one can derive the expressions for j2 and j3 by introduction of diffusion occupancy factors 1

F2,ois

  1 1  6   ,  X 1 1  X 3  X 3 1  X 1  

F3,ois

  1 1  6   . X 1  X X 1  X     1 2 2 1  

(12.2)

1

(12.3)

Using the diffusion tensor inclusive the elastodiffusion tensor, Eq. (1), instead of the scalar diffusion coefficient D in Eq. (11), the final expression for the three dimensional diffusion equation of interstitials in ois reads as 5

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* D 3 el    D   ij ij  x . RgT  j 1 j

ji   Fi , ois

(13)

Note that for the stress free dilute system Fi , ois  X and Dijel  0 . Thus, the site fraction X in Eq. (10) plays the role of the occupancy factor. Eq. (13) is then a natural extension of the traditional diffusion equation, Eq. (10).

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3.2. The tis-case

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The diffusion paths for tis in the 1-direction are not straightforward (they are branched) as visible in Fig. 1b. The atoms pass the sequence of tis of types 1, 2, 1, 3, 1, 2, 1, 3, 1,…. Thus, diffusion in the 1-direction consists of four steps 1  2  1  3  1 . The respective mean times t1,12 , t1,21 , t1,13 and t1,31 are given similarly to Eqs. (4) and (5). In analogy to ois, the flux in the 1-direction can be expressed as 1

  1 1 1 1 * j1   Atis       x1 . X 1  X X 1  X X 1  X X 1  X         2 2 1 1 3 3 1   1

(14)

One can evaluate the pre-multiplier Atis as Atis  12D  RgT   . The respective diffusion

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occupancy factors read then as

1

F2,tis

  1 1 1 1  12      , X 1  X X 1  X X 1  X X 1  X         1 2 2 1 2 3 3 2  

F3,tis

  1 1 1 1  12      . X 1  X X 1  X X 1  X X 1  X         1 3 3 1 2 3 3 2  

(15.1) 1

(15.2)

1

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F1,tis

  1 1 1 1  12      , X 1  X X 1  X X 1  X X 1  X         1 2 2 1 1 3 3 1  

(15.3)

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The final diffusion equation is given in analogy to Eq. (13) as ji   Fi ,tis

* D 3 el    D    ij ij x . RgT  j 1 j

(16)

3.3. Complete treatment of interstitial diffusion in stressed bcc crystals First of all it shall be recapitulated that an adapted equation for the diffusive flux of an interstitial, Eq. (13) or (16), has been derived. The diffusive flux depends on both the diffusion path with differently occupied sites, expressed by occupancy factors Fi , and on the 6

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deformation/stress state via the elastodiffusion tensor DDijel . To obtain equations for the system evolution, Eqs. (13) and (16) must be supplemented by the mass conservation law (see Eq. (8) in [27]), the formulation of the generalized chemical potential  * (see Eq. (6) in [27]) and the rule for distribution of the atoms amongst individual types of sites (see Eqs. (9-13) in [27]). Finally it shall be mentioned that the former diffusion equation, Eq. (7), in [27], includes already a diffusion occupancy factor, however, derived in a much simpler phenomenological way without accounting for the lattice structure. Consequently the previous diffusion occupancy factor does not provide any additional anisotropy effect. Using Eq. (13) or (16) of the current paper instead of Eq. (7) in [27] completes now the diffusion theory and provides a physically sound concept for the quantitative treatment of interstitial diffusion in stressed crystals. 4. Example

F1,ois

For 3 X1  3 X 2  3 X 3  X  1 , the values of diffusion occupancy factors are  F2,ois  F3,ois  F1,tis  F2,tis  F3,tis  X . To demonstrate the significant dependence of the

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diffusion occupancy factors on X1 , X 2 , X 3 , they are plotted in Fig. 2 for a fixed X in dependence on X 1 X by assuming X 2  X 3 . If the mechanical part of the generalized

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chemical potential significantly exceeds RgT , i.e. for stresses of the magnitude of several GPas (as typical for the vicinity of a crack tip or a dislocation core), the individual values of X i may vary nearly in the range of 0 to 1. Then, as evident from Fig. 2, the diffusion occupancy factors may assume rather low values, and the diffusion process is blocked. For X1  0 the diffusion remains unblocked only in 1-direction for ois. For X1  1, diffusion is

F1,ois

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0

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blocked in all directions for both ois and tis. In the case of two site fractions X i  0 (all atoms are concentrated in the third complementing type of ois or tis), diffusion is blocked in all directions for ois or tis. 0

10

F1,ois

F1,ois

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F/X

-1

10

F/X

-1

10

F1,ois

F2,ois, F3,ois, F1,tis

-2

10

F2,ois, F3,ois, F1,tis -2

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F2,tis,F3,tis

F2,tis,F3,tis -3

-3

10

10

0.0

0.2

0.4

0.6

X1/X

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

X1/X

a) b) Fig. 2. Dependence of diffusion occupancy factors F1,ois , F2,ois , F3,ois , F1,tis , F2,tis , F3,tis , normalized with X , on X 1 X by assuming X 2  X 3 for a) X  0.01 and b) X  1 .

5. Comment

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ACCEPTED MANUSCRIPT The authors are aware of the limitations of their model given by the following assumptions: i. ii.

the interstitial atoms occupy either ois or tis. In reality both ois and tis are occupied with different probabilities. diffusion occurs exclusively by jumping between either the nearest ois or the nearest tis. In reality diffusion may occur by utilizing both ois and tis.

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If, however, the occupancy energies of ois and tis by an interstitial are significantly different (this is the case for C and H in bcc Fe) and the temperature is sufficiently low, our assumptions meet the reality well. Since the diffusion occupancy factors may assume very low values, the model demonstrates that a high-level stress field may significantly decelerate the diffusion process. 6. Conclusion

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Acknowledgement

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The role of impurity interstitials for H-embrittlement or formation of Cottrell clouds of C-atoms near dislocation cores is still of significant industrial relevance. The understanding of the physical background is still an increasing demand on the reliability of modern steels. Consequently modeling and simulation of the diffusion process of interstitial atoms as the controlling process are more and more involved in research and practice. To allow realistic predictions by simulations, the proper description of the diffusion process by reliable equations is an according necessity. However, up to now the actual paths of the diffusing interstitial atoms have not been implemented in the description of the diffusion process. The diffusion concept presented in the current work takes now into account the actual paths as well as the different occupancies of interstitial sites in a stressed lattice. The newly developed equations of the diffusive fluxes exhibit now a remarkable anisotropy of the diffusion process. The provided model seems more efficient and appropriate than atomistic approaches as molecular dynamics, see e.g., [30], or Monte Carlo simulations.

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Financial support by the Austrian Federal government (in particular from the Bundesministerium für Verkehr, Innovation and Technologie and the Bundesministerium für Wirtschaft und Arbeit) and the Styrian Provincial Government, represented by Österreichische Forschungsförderungsgesellschaft mbH and by Steirische Wirtschaftsförderungsgesellschaft mbH, within the research activities of the K2 Competence Centre on “Integrated Research in Materials, Processing and Product Engineering”, operated by the Materials Center Leoben Forschung GmbH in the framework of the Austrian COMET Competence Centre Programme, Projects A1.23 and A2.32, is gratefully acknowledged. J.S. gratefully acknowledges the financial support by the Czech Science Foundation in the frame of the Project 17-01641S.

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