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Effects of alloying elements on vacancies and vacancy-hydrogen clusters at coherent twin boundaries in nickel alloys
Accepted Manuscript Effects of alloying elements on vacancies and vacancy-hydrogen clusters at coherent twin boundaries in nickel alloys Xiao Zhou, Jun Song PII:
S1359-6454(18)30068-5
DOI:
10.1016/j.actamat.2018.01.037
Reference:
AM 14331
To appear in:
Acta Materialia
Received Date: 10 November 2017 Revised Date:
29 December 2017
Accepted Date: 2 January 2018
Please cite this article as: X. Zhou, J. Song, Effects of alloying elements on vacancies and vacancyhydrogen clusters at coherent twin boundaries in nickel alloys, Acta Materialia (2018), doi: 10.1016/ j.actamat.2018.01.037. 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 proof before it is published in its final 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.
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Effects of Alloying Elements on Vacancies and Vacancy-Hydrogen
Xiao Zhou1, Jun Song1,* 1.
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Clusters at Coherent Twin Boundaries in Nickel Alloys
Department of Mining and Materials Engineering,
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Abstract
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McGill University, Montréal, Québec, H3A 0C5, Canada
The interactions between several typical alloying elements (i.e., Al, Cr, Mo, Nb and Ti) and hydrogen at coherent twin boundaries (CTB) in Ni alloys were systematically studied through first-principles calculations. It was found that solute atoms generally prefer to segregate at CTBs, with the segregation tendency prescribed by the size
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mismatch between the solute and host Ni atom. Moreover, there exist attractive interaction between certain solute atoms, potentially promoting solute co-segregation at CTBs. Though the solute presence does not favor hydrogen accumulation, it can considerably reduce the formation energies of vacancies and vacancy-hydrogen clusters
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at CTBs, which, if augmented by large-scale plastic deformation, may facilitate nanovoid
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nucleation to promote eventual crack initiation at CTBs. The present study demonstrated the critical role of alloying in hydrogen-induced crack initiation at CTBs, providing a new perspective towards understanding HE in Ni alloys.
Keywords: Hydrogen emrbittlement; coherent twin boundary; solute segregation; vacancy-hydrogen cluster; nanovoid nucleation. *
Corresponding author. E-mail address: [email protected] (Jun Song)
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1. Introduction Hydrogen-induced degradation of materials has been a subject of continuous interest
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for many decades [1-3]. In particular, the degradation by hydrogen embrittlement (HE) [4-6], which causes many high-strength metals, such as high strength steels and Ni alloys,
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to undergo premature failure with a drastic loss in ductility, toughness and strength [2, 7, 8], has been a long-standing industrial nuisance. Albeit enormous research efforts, there is
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still no general theory, nor consensus on the mechanisms underlying HE [5, 9-13]. HE in polycrystalline metals is further complicated, and often manifested, by the presence of microstructures, such as grain boundaries (GBs), which affect hydrogen diffusion and segregation, and modify the local lattice cohesion and deformation behaviors[14-16].
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Bechtle et al. [17] demonstrated the paramount importance of GBs in controlling the material’s susceptibility to HE.
They showed that the severity of HE of pure Ni can be
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significantly reduced by engineering the microstructure to enhance the fraction of special GBs that consist of principally Σ3 twin boundaries. Among the various special GBs, the
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coherent twin boundary (CTB) is of particular interest. With nearly identical atom arrangement as the bulk lattice, CTBs are often regarded to be highly resistant to fracture and HE [18, 19]. Nonetheless, contradicting results regarding the role of CTBs exist in the literature. In the study by Ulmer and Altstetter [20] on hydrogen-assisted fracture in austenitic stainless steels, it was found that cracking frequencies at annealing twins and other GBs were approximately the same. Seita et al. [21, 22] performed in situ tensile 2
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tests on hydrogen charged Ni-based superalloy samples and identified CTBs as the preferential microstructural sites for crack initiation.
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The above conflicting opinions about CTBs are intriguing and suggest the necessity of further studies. One puzzle piece absent from those previous studies is the effect of alloying elements (thereafter also referred to as solute atoms below). With alloying being
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commonly used to modulate the mechanical properties and corrosion resistance of Ni
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alloys, solutes are unavoidably present in the Ni lattice [23-25]. As reported by many previous studies [26-28], solute atoms may segregate at GBs. Therefore understanding of HE at CTBs necessitates the investigation of solute-hydrogen interplay at CTBs. In this regard, here we study the role of typical solute atoms (i.e., Al, Cr, Mo, Nb and
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Ti) in affecting HE at CTBs in Ni alloys, employing first-principles calculations. Their segregation and co-segregation at CTBs, and interactions with hydrogen were examined.
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Meanwhile, a few recent studies on Ni-based alloys [29, 30] suggest that microvoids serve as nucleation sites for crack initiation sites along GBs in hydrogen-charged samples.
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Consequently we investigated the coupled effect of solute and hydrogen on vacancy accumulation and subsequently nanovoid nucleation at CTBs. In the end, the implication of solute presence to hydrogen-induced CTB cracking and HE susceptibility of Ni alloys was discussed.
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2. Computational Methodology First-principles calculations on the basis of density functional theory (DFT) using the
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Vienna ab-initio simulation package (VASP) [31, 32] were performed to examine the interplay between alloying elements with vacancies and vacancy-hydrogen clusters.
atoms.
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Here a vacancy-hydrogen cluster refers to a vacancy bound with one or several hydrogen For simplicity, below we denote the vacancy-hydrogen cluster as VHx with x In the calculations, the electron-ion
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being the number of hydrogen atoms in the cluster.
interactions were described by Blöch projector-augmented wave method (PAW) [33, 34], and Generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [35] based on plane-wave basis sets was employed for electron exchange and correlation For calculations involving a bulk lattice, a 3×3×3 periodic supercell
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functionals.
containing 108 atoms was used, with a k-point grid of 5×5×5 and the plane-wave cut-off
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energy being 400 eV, while for calculations involving CTB, a periodic supercell model of dimensions 8.8Å×5.0Å×24.4Å containing 96 atoms, constructed based on coincidence
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site lattice theory (CSL) [36] was used with a k-point grid of 5×5×10 and a cutoff energy of 380 eV.
In all calculations, the convergence criteria of energy and atomic force were
set as 10-5 eV and 10-2 eV/Å respectively, and the first-order Methfessel-Paxton scheme with a 0.1 eV smearing width was adopted. The supercell of CTB used in this study is schematically illustrated in Fig. 1.
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The alloying elements were introduced into the Ni lattice by substituting host Ni atoms. The alloy elements considered in our study are Al, Cr, Mo, Nb and Ti, which are
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widely used in commercial Ni-based alloys (e.g., Inconel 725 [21]) for improving the mechanical properties. The solution energy of an isolated alloying element, denoted as ES , is defined as
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ES = Etot [ Nin -1S ] − Etot [ Nin ] − ES0 + E Ni0 ,
(1)
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where Etot [Ni n ] is the total energy of the reference bulk or CTB Ni system, Etot [Ni n −1S ] is the total energy of the corresponding bulk or CTB Ni system where one Ni atom is substituted by an alloying element, ES0 is the cohesive energy of the alloying atom in its stable crystalline state and ENi0 is the cohesive energy of Ni. Apparently the solution
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energies of a solute in the bulk and at CTB may differ.
The difference can be
characterized the segregation energy, ESseg , defined as (2)
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ESseg = ESCTB − ESBulk
where ESCTB and ESBulk denote the solution energies of the solute located at CTB and in
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the bulk respectively. ESseg effectively indicates the tendency of the solute to segregate at CTB.
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Fig. 1 (Color online): Schematic illustration of the CTB model used in DFT calculations in this study. The green and grey spheres indicated atoms at the CTB and in the bulk respectively. The circle with dash line represented the vacancy and the small red balls indicated the positions of hydrogen atom. The left and right figures above are projection views the system along [110] and [111] directions respectively.
The interaction between hydrogen and a solute atom S, EbSH , is defined as EbSH = E [ Ni n −1 SH ] − E [ Ni n −1 S ] − E [ Ni n H ] + E [ Ni n ] ,
(3a)
where E [ Ni n −1SH ] is the total energy of a (bulk or CTB) Ni lattice containing n-1 Ni
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atoms, one solute atom S and one hydrogen, E [ Ni n −1S ] and E [ Ni n H ] are the total energies of a lattice containing n-1 Ni atoms plus one solute atom S and a lattice that
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holds n Ni atoms plus one hydrogen atom respectively, and E [ Ni n ] is the total energy
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of the reference Ni lattice. Eq. 3a can also be generalized for the situation of multiple hydrogen adsorption, as the following: EbSH = E [ Ni n−1 SH m ] − E [ Ni n−1 SH m −1 ] − E [ Ni n H ] + E [ Ni n ] ,
(3b)
Meanwhile the interaction between solute atoms can be analyzed in a similar way, with the binding energy Ebαβ between two solute atoms, α and β , in the Ni lattice defined as:
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Ebαβ = E [ Ni n − 2αβ ] − E [ Ni n −1α ] − E [ Ni n −1 β ] + E [ Ni n ] ,
(4)
where E [ Ni n − 2αβ ] is the total energy of a (bulk or CTB) Ni lattice containing n-2 Ni
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atoms and two solute atoms α and β , and E [ Ni n −1 x ] ( x = α or β ) is the total energy of a Ni lattice containing n-1 Ni atoms and one solute atom x.
In our definitions (i.e.,
Eqs 3-4 above), positive and negative values of the binding energy indicates repulsion
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and attraction respectively between hydrogen and the solute atom, or two solute atoms.
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The vacancy was created by removing a host Ni while a vacancy hydrogen cluster VHx is constructed by introducing hydrogen atoms into interstitial sites neighboring the vacancy. For a vacancy (essentially VHx with x = 0) or VHx, the formation energy E VH f is defined as VH x
= E [ Ni n −1 , VH x ] + ENi0 − E [ Ni n ] −
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Ef
x
2
EH0 ,
x
(5)
2
where E [ Ni n −1 , VH x ] is the total energy of a system containing n-1 host Ni atoms and 1
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VHx cluster, E [ Ni n ] is the total energy of the corresponding Ni lattice that contains n host Ni atoms. The above formation energy of VHx, in the presence of solute atoms, is
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modified and becomes
E f , sol = E [ Ni n − m −1S m , VH x ] + ENi0 − E [ Ni n − m S m ] − VH x
x 2
EH0 ,
(6)
2
where E [ Ni n − m −1S m , VH x ] is the total energy of the system containing n-m-1 host Ni atoms, m solute atoms, and one VHx cluster, E [ Ni n − m S m ] is the total energy of the system containing n-m host Ni atoms and m solute atoms. The difference between E VH and f x
E VH thus provides a metric to assess the influence of solute on the formation of VHx: f , sol x
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VH x x x ∆E VH = E VH . f f , sol − E f
(7)
Note that for the VHx cluster, only hydrogen segregation at the nearest-neighboring Meanwhile in
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octahedral sites surrounding the vacancy was considered, and thus x≤6.
our investigation of solute-vacancy and solute-VHx interactions, we only consider the situation of solute atoms residing in the immediate vicinity (i.e., at nearest neighboring In addition, one thing worth noting in the above
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sites) of the vacancy or VHx.
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calculations is that there can be multiple configurations for a particular solute-hydrogen, solute-solute, solute-vacancy or solute-VHx interaction, and we only considered the configuration of the lowest energy.
Meanwhile we also examined the segregation tendency of VHx towards CTB.
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x (CTB) and Denoting the formation energies of VHx at CTB and in bulk Ni as E VH f
x (Bulk) E VH respectively, we can define the segregation energy of VHx, in a similar way as f
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that of the segregation of solute atom:
VH Eseg = E VH f x
x
( CTB)
− E VH f
x
( Bulk)
(8)
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In the case where CTB has pre-segregated solute atoms, the segregation energy, now VH x denoted as Eseg , sol , becomes VH VH ( CTB) Eseg − E VH ,sol = E f ,sol f x
x
x
( Bulk)
,
(9)
( CTB) represents the formation energy of VHx at CTB, affected by the where E VH f ,sol x
VH x x ( CTB) presence of solute. One fact to note is that E VH and Eseg are related to each f , sol
other via ∆E VH (at CTB), a term previously defined in Eq. 7, which can be expressed as f x
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VH x VH x VH x Eseg , sol − Eseg = ∆E f
(10)
3.1 Solute atoms and VHx in bulk Ni
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3. Results and discussion
We first examined the energetics of the solute atom, VHx and their mutual
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interactions in the bulk Ni lattice, in order to establish a baseline for our subsequent study
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of the CTB.
3.1.1 Solute-solute and solute-hydrogen interactions in bulk Ni
Considering two nearest-neighboring solute atoms, we calculated the corresponding binding energies, shown in Fig. 2. Almost exclusively for all solute pairs (except for the
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case of Cr-Cr which shows a tiny negative Eb of -0.03 eV), the binding energy exhibits a positive value, indicating that solute atoms do not want to be in the immediate neighbors
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of each other, and thus no tendency for solute atoms to aggregate into clusters in the bulk Ni lattice. The binding energies of solute-hydrogen interactions (with hydrogen located
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at an interstitial site immediately neighboring the solute atom) are shown in Fig. 2b, again showing positive values, suggesting a repulsive interaction between solute and hydrogen atoms.
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Fig. 2: Binding energies of different (a) solute-solute ( , see Eq. 3a) couples in bulk Ni.
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(
, see Eq. 4) and (b) solute-hydrogen
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3.1.2 Effect of solute atoms on the formation of VHx in bulk Ni
The formation energies of VHx in a solute-free bulk Ni lattice are listed in Table I.
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The results are in good agreement with previous studies [37-41]. Fig. 3 illustrates the effect of solute atoms (i.e., ∆E VH , see Eq. 7) on the formation of VHx. Note that here f x
we only considered the presence of individual solute atoms in the immediate vicinity of a VHx cluster, because of the repulsion between solute atoms (cf. Fig. 2a). In addition, it is worth noting that only those nearest octahedral interstitial sites neighboring the vacancy
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are considered for potential hydrogen trapping and consequently the maximum number of
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trapped hydrogen is six.
Table I: The formation energies of vacancy (i.e., VHx with x=0) and vacancy hydrogen clusters (i.e., VHx with x=1-6) in pure Ni, in bulk lattice (
Data from literature 1.48 , 1.40[38], 1.14[39], 1.67[40], 1.44[41] 1.35[37], 1.21[38] 1.21[37], 1.01[38] 1.07[37], 0.87[38] 0.93[37], 0.70[38] 0.79[37], 0.57[38] 0.65[37], 0.33[38] [37]
(
)
).
x (CTB) E VH (eV) f
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This work 1.40 1.26 1.10 0.98 0.85 0.72 0.57
) and at CTB (
This work 1.42 1.28 1.05 0.94 0.80 0.67 0.53
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0 1 2 3 4 5 6
x (Bulk) E VH (eV) f
)
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Number of hydrogen
(
As can be seen in Fig. 3, for x between 0 and 4, the formation of VHx is favored by <0) while impeded by the existence of Cr and Mo the presence of Al, Nb or Ti ( ∆E VH f
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x
( ∆E VH >0). In the cases of x=5 and 6, however, all solute atoms show positive ∆E VH f f x
x
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values, resisting the formation of VHx.
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Fig. 3 (Color online): The effect of solute atoms, reflected by ∆
(see Eq. 7), on the
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formation of vacancy hydrogen cluster VHx as x (i.e., Number of H) varies.
3.2 Solute atoms and VHx at CTB
3.2.1 Segregation of solute atoms at CTB
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For the case of CTB, we first examined the segregation tendency of solute atoms. As seen in Fig. 4, all solute atoms exhibit negative ESseg , indicative of it being energetically preferable for solutes to segregate at CTB. Fig. 4 also indicates that those solute atoms
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have different segregation tendency, being strongest ( ESseg = -0.10 eV) for Nb and
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weakest for Cr ( ESseg = -0.05 eV).
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Fig. 4 (Color online): (Top) DFT calculated and continuum predicted (see Eq. 13) segregation energies of different solute atoms at CTB. (Bottom) The elastic energy and size factor Φ corresponding to different solute atoms. The lines are drawn to guide the eye.
The difference can be attributed to that the energetics of substitutional solute atoms are greatly influenced by the size mismatch between the solute and host atoms, as well
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acknowledged by many previous studies [42-44]. Denoting V0 and Vs as the atomic
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volume of a host Ni atom and solute atom respectively, the size mismatch can be quantified via the size factor ( Φ ): Φ=
Vs − V0 V0
.
(11)
Note that the V0 of a host Ni atom can be approximated by its Voronoi volume [45], being different for Ni atoms in the bulk and at CTB. The size mismatch leads to lattice distortion and subsequently gives rise to an elastic energy Eel, which can be analyzed 13
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within the framework of continuum mechanics as proposed by Eshelby [42, 46], treating the solute substitution process as the insertion of a spherical inclusion of volume Vs into a
Eel =
Bs ( ∆Vs ) 2 2Vs
+
2G0 (∆V0 ) 2 3V0
.
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hole of volume V0 in the host matrix : (12)
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where ∆Vs and ∆V0 are the volume changes of spherical inclusion and hole due to the internal stress, Bs is the bulk modulus of inclusion, and G0 is the shear modulus of host
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matrix. Considering the stress continuity at the inclusion-matrix interface, the above can be rewritten as [42]:
Eel =
2 Bs G0 (Vs − V0 )2 . 3BsV0 + 4G0Vs
(13)
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With V0 being different for the bulk and CTB, Eel thus exhibits different values (denoted as E elB and EelCTB below) for a substitutional solute in the bulk and CTB, and
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el the difference, Eseg (see Eq. 14 below), effectively defines the segregation energy
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predicted from continuum mechanics: el Eseg = EelCTB − EelB .
(14)
The values of size factor Φ , elastic energy EelCTB and continuum-predicted segregation el el energy Eseg of solute atoms at CTB are all presented in Fig. 4. We note that Eseg
exhibits essentially the same trend as the DFT-calculated segregation energy ESseg , albeit some difference in the absolute value. This observation suggests that the mechanical
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energy be the dominant factor in determining the segregation tendency of solute atoms towards CTB.
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The preferential segregation will certainly lead to enrichment of solute atoms at CTB. To have an idea of the level of solute enrichment at CTB, we consider two Ni alloys, Inconel 718 and 725, as representatives, and the concentration ( CSCTB ) of each alloy
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element at CTB is roughly estimated based on the following equation:
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CSCTB
− E seg CSBulk exp S k BT , = − ESseg Bulk 1 + CS exp k BT
(15)
where C SBulk is solute concentration in the bulk lattice and kB is Boltzmann constant and
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T is temperature. CSBulk is taken as either the alloy element solubility (obtained from FactSage alloying database [47, 48]) or alloy composition [22, 49], whichever is lower. Using the segregation energies obtained above (see Fig. 4), the estimated atomic
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concentrations of solute atoms (at room temperature) expected at the CTB are listed in
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Table II, from which we can see that in equilibrium appreciable amount of Cr, Ti, Mo and Nb would be present at CTB. It is worth to note that the above calculation is a rough estimation without accounting for the mutual interaction and competition between different solute elements. As a result, the CSCTB value obtained from Eq. 15 serves more of an indication of the solute segregation tendency, and can differ from the actual solute
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concentration at CTBs in those Ni alloys (which would depends on the complex interplay between solutes and the processing conditions).
Element
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Table II: Bulk solute concentration and corresponding solute segregation expected at CTB for two representative Ni alloys, Inconel 718 and 725. Inconel 718
Inconel 725
Atomic concentration (at. %)
Atomic concentration (at. %)
CTB
Bulk†
Cr
4.24
22.69
4.24
Ti
1.35
14.58
Nb
2.76
54.44
Mo
0.51
13.39
Al
0.76
7.12
CTB
22.69
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Bulk
1.97
19.94
2.41
51.07
0.51
13.39
0.65
6.16
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† Note that the bulk concentration is approximately taken as either the alloy solubility (at 300K) or alloy composition, whichever is lower.
3.2.2 Solute-solute and solute-H interactions at CTB
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The appreciable segregation of solute atoms would unavoidably lead to mutual interaction between solute atoms. Fig. 5a shows the binding energies of different solute
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pairs. Unlike the case in bulk Ni (cf. Fig. 2), Ebαβ exhibits negative values for several pairs (with EbMoNb = -0.25 eV being the lowest, see Fig. 5a), indicating the possibility of
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solute co-segregation at CTB.
The interaction between hydrogen and individual solute atoms at CTB is
demonstrated in Fig. 5b. Here in our calculation (up to six) hydrogen atoms were introduced to the interstitial sites immediately neighboring a solute atom at CTB, and the most energetically stable configurations were determined, from which the binding energy
EbSH was then obtained. As seen in Fig. 5b, EbSH generally exhibits positive values, 16
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indicative of repulsion between hydrogen and individual solute atoms. The repulsion becomes stronger as the number of H increases. Meanwhile, the interaction between
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hydrogen and solute pairs was also examined and EbSH was also found to remain positive, as shown in Fig. 6. The above results demonstrate that solute segregation alone
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would impede hydrogen accumulation at CTB.
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Fig. 5 (Color online): (a) Binding energies ( Binding energies (
, see Eq. 4) of different solute pairs at CTB. (b)
, see Eqs 3a-b) between hydrogen and individual solute atoms at CTB.
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Fig. 6: Binding energies ( at CTB.
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, see Eq. 3a) between one hydrogen atom and different solute pairs
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3.2.3 Effect of solute atoms on the formation of VHx at CTB
Fig. 7: The effect of (a) individual solute atoms and (b) solute pairs, reflected by ∆
(see Eq.
7), on the formation energy of a single vacancy (i.e., VHx with x = 0) at CTB.
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The formation energies of VHx at CTB in pure Ni are listed in Table I. As seen in the table, they show negligible difference in comparison to those in pure bulk Ni. Below
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we elaborate on how the formation of VHx at CTB is affected by the presence of solute. For a single vacancy (i.e., VH0) at CTB, the influence of individual solute atoms and solute pairs on its formation is shown in Fig. 7. Note for solute pairs we only consider
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those with tendency of co-segregation (i.e., Ebαβ being negative, see Fig. 6 above) at
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CTB. For the case of individual solutes, we note from Fig. 7a that while Mo and Cr increase the vacancy formation energy, Al, Ti and Nb reduce the vacancy formation energy with Nb leading to the largest reduction of 0.15 eV. For the case of solute pairs, the Cr-Mo increases the vacancy formation but other pairs considerably lower the
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vacancy formation energy. In particular, we see (cf. Fig. 7b) that the Nb-Ti pair has the most significant effect, reducing the vacancy formation energy by 0.26 eV (~18%
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reduction). Overall the presence of solutes and solute pairs predominantly tends to reduce
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the vacancy formation energy, facilitating vacancy formation at CTB.
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∆
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Fig. 8 (Color online): The effect of (a) individual solute atoms and (b) solute pairs, reflected by (see Eq. 7), on the formation energies of vacancy hydrogen clusters VHx at CTB.
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With hydrogen present, the effect of solute and solute pairs on the formation of VHx is illustrated in Fig. 8. For individual solutes (cf. Fig. 8a), we note that Nb and Ti, and Al
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can effectively reduce the formation energy of VHx with x up to 4 and 2 respectively. Meanwhile Cr can also reduce the formation energy of VHx with x ≤ 3, with the effect being much less pronounced. In the contrary, the presence of Mo always inhibits VHx x formation. For VHx with x ≥ 5, all solute atoms lead to positive ∆E VH , resisting the f
formation of VHx.
For solute pairs (cf. Fig. 8b), we see that Mo-Nb (slightly), Al-Nb
and Nb-Ti favor the formation of VHx with x = 1, x ≤ 2 and x ≤ 3 respectively, while 20
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otherwise the formation of VHx is impeded by the presence of solute pairs.
Overall,
from Fig. 8, we see that the presence of solute (individual or pair) can greatly inhibit
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excessive hydrogen aggregation at vacancies but may promote the formation of “light” vacancy-hydrogen bundles, in particular VH1 and VH2.
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3.6 Synergetic effect of hydrogen and solute under plastic deformation
In previous sections, we have demonstrated that the formation of vacancies and VHx
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clusters can be notably affected by the presence of solutes, and in certain cases greatly facilitated. Normally the vacancy concentration at ambient temperature is very low in annealed structural metal samples (e.g., ~10-23 in Ni [37]), in which case hydrogen and
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solutes would play limited role in modifying the mechanical behaviors of metals. However, excess vacancies can be induced by plastic deformation. For instance, Zehetbauer et al. [50, 51] demonstrated that the concentration of vacancies produced in
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Ni and Cu by severe plastic deformation at room temperature can reach the order of 10-4
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or higher, close to the thermal equilibrium concentration of vacancies at the melting temperature. As suggested by Militzer et al. [52], the net rate of generation of plasticity-induced excess vacancies can be expressed as:
dCV = Γ −ξ , dt
(16)
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where CV denotes the concentration of excess vacancies, and Γ and ξ are the production and annihilation rate respectively. The production rate Γ can be expressed
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as [52]: (17)
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where χ is a dimensionless constant, being approximately 0.1 [53, 54], Ω 0 is the
Q f is the vacancy formation energy.
is the strain rate and
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atomic volume (being 1.09×10-29 m3 for Ni), σ is the stress,
In the presence of hydrogen atoms, those plasticity-induced vacancies may be stabilized to remain in the lattice during the deformation process. As suggested by Li et al.
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[13], under the condition that the concentration of hydrogen exceeding vacancy population CV, most of the vacancies will combine with hydrogen atoms to form VHx clusters, which prevents them from easy annihilation and effectively renders ξ zero as a
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first-order approximation, consequently leading to
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(18)
Note that in this equation CV includes both vacancies and VHx clusters. By integrating the above equation, one linear relation between CV and ε can be obtained:
CV =
χΩ 0σ Qf
ε + C0 ,
(19)
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where C0 is the thermal equilibrium vacancy concentration in Ni bulk lattice expected in absence of plastic deformation (~10-23 at room temperature [37]).
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From Eq. 19, the bulk concentration of plasticity induced excess vacancies CV in presence of hydrogen can be determined as a function of σ and ε .
As an example,
Fig. 9 shows the mapping of CV (being the ratio of vacancies over Ni atoms) for Inconel
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725 for the ranges of stress/strain of typical stress-strain curves under hydrogen charging
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at room temperature [21].
Fig. 9 (Color online): The plastic deformation induced vacancy concentration in bulk Ni lattice, Cv according to Eq. 17, within the typical stress-strain range for Inconel 725 [21].
As seen from Fig. 9, the plastic deformation result in CV to range from 10-4 to 10-3, being significantly higher than the normal vacancy concentration (~10-23) expected at
23
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room temperature in absence of plasticity. This substantial bulk presence of vacancies will be further manifested at CTB and affected by solute atoms, as suggested by our
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previous results. Denoting the concentration of vacancies (including VHx) at CTB as CVCTB , it can be roughly estimated as
(20)
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CVCTB
VH x − Eseg , sol CV exp k T B CTB + C CTB , = CS × V0 VH x − Eseg , sol 1 + CV exp k T B
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VH x where CSCTB denotes the solute or solute pair concentration at CTB, Eseg , sol (cf. Eqs.
9-10) denote the segregation energy of vacancy-hydrogen cluster at CTB in the presence of solute. Note here for simplicity we assume pre-segregation of solute atoms at CTB
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before the segregation of vacancy-hydrogen clusters occurs. The term CVCTB denotes the 0 total vacancy concentration expected at CTB in absence of solute, related to CV as:
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CVCTB 0
− E VH x CV exp seg kT B , = CV × VH x − Eseg 1 + CV exp k T B
(21)
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VH x with Eseg previously defined in Eq. 8.
As previously elaborated (in Section 3.2.3), certain solute atoms (i.e., Cr, Ti, Nb and
Al) or solute pairs (i.e., Nb-Ti, Al-Nb, Al-Ti and Cr-Nb) can effectively reduce the formation energy of single vacancies and/or VHx (for some x), therefore possibly contributing to enhancing the total vacancy population. Fig. 10 and Fig. 11 shows the calculated CVCTB under different individual solute (i.e. C SCTB ) and solute pair 24
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concentration levels, as a function of CV. We note that the total vacancy population at CTB can be substantially elevated by the solute presence, being particularly significant in
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the presence of Nb, Al, Nb-Ti and Al-Nb. These results suggest that under plastic deformation, the synergetic effect of hydrogen and solute may promote considerable
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vacancy aggregation at CTBs.
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Fig. 10 (Color online): Total vacancy population at CTB ( ), as a function of bulk vacancy concentration (CV), influenced by the presence of different levels of individual solutes at CTB.
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Fig. 11 (Color online): Total vacancy population at CTB ( ), as a function of bulk vacancy concentration (CV), influenced by the presence of different levels of solute pairs at CTB.
One consequence that naturally follows the enhanced vacancy concentration at CTB is the increasing possibility of vacancy coalescence, which would subsequently lead to
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nanovoid nucleation. Previous studies have well demonstrated the critical relation
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between vacancy concentration and (nano)void nucleation [30, 55]. Reina et al. [56] formulated a model of nanovoid nucleation under severe plastic deformation, relating the nucleation to diffusion-mediated vacancy aggregation and subsequent vacancy coalescence. In their study, it was postulated that a certain time tf is required to establish a steady-state concentration CV of vacancies:
tf =
CV
v0 exp − ( EVf + E if ) / kBT
,
(22)
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where v0 is the Debye frequency, and
and
!
denote the formation energies of a
vacancy and far-field self-interstitial respectively. The time tf can be regarded as a lower
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bound of the time required for the nanovoid nucleation. Below we perform a simple analysis to assess the effect of solute on tf, thus roughly gauging its influence on the kinetics of nanovoid nucleation.
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Applying the above equation to CTB, we can obtain the required times to achieve a threshold concentration of solute atoms, respectively as: t
=
CTB f
CVCTB
,
(23a)
CVCTB
,
(23b)
v0 exp − ( EVf + E if ) / k BT
v0 exp − ( E Vf , s + E if ) / k BT
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t CTB f ,s =
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of vacancies at CTB, without and with the presence of
from which we may define a normalized time parameter as:
t CTB f
= exp ∆E f / k BT ,
(24)
is essentially ∆E VH defined previously in Eq. 7. We see that tn is dependent f x
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where Δ
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tn =
t CTB f ,s
on the solute-induced change in the vacancy formation energy. As illustrated in Fig. 12, certain solutes or solute pairs can render tn << 1 for vacancy and vacancy-hydrogen clusters. This indicates that the presence of solute atoms may contribute to decreasing the equilibration time required for attaining a particular stead-state vacancy concentration at CTBs, thereby facilitating nanovoid nucleation. The accelerated nanovoid nucleation,
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together with possible localization of dislocation activities at CTBs [57], eventually may assist or promote crack initiation. Furthermore, the above scenario would also be
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generally applicable to other GBs.
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Fig. 12 (Color online) The normalized time tn (see Eq. 24, at T=300K) corresponding to single vacancy (VH0) and vacancy-hydrogen clusters (VH1 and VH2), influenced by the presence of individual solutes (top) and solute pairs (bottom) at CTBs. The lines are drawn to guide the eye.
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4. Conclusions
In summary, comprehensive first-principles calculations have been performed to
investigate the interplay between hydrogen and typical alloying elements (i.e., Al, Cr, Mo, Nb and Ti) in Ni alloys, and their coupled effect on the formation of vacancy and vacancy-hydrogen clusters at coherent twin boundaries (CTBs) in Ni alloys. The key findings are listed as the follows:
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1) It is energetically preferable for solute atoms to segregate at CTBs, with the segregation tendency directly prescribed by the size mismatch between the solute
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and host Ni atom; 2) Unlike the case of bulk Ni lattice where different solute atoms repel each other, at CTBs there exists attractive interaction between certain solute atoms, potentially
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promoting co-segregation;
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3) The solute presence at CTB generally inhibits hydrogen accumulation, but can considerably reduce the formation energies of vacancies and vacancy-hydrogen clusters;
4) This considerable reduction in the formation energy, when augmented by large-scale
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plastic deformation, may lead to marked increase of total vacancy population at CTBs, and subsequently facilitate nanovoid nucleation to aid eventual crack
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initiation at CTBs.
The present study elucidates possible implication of alloying to hydrogen-induced CTB
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cracking, providing a new perspective towards understanding HE in Ni alloys.
Acknowledgements XZ and JS would like to acknowledge financial support by Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant # RGPIN
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418469-2012), National Natural Science Foundation of China (NSFC Grant No. 51628101) and China Scholarship Council (CSC), and thank Supercomputer Consortium
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Laval UQAM McGill and Eastern Quebec (CLUMEQ) for providing computing resources.
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33
S1359-6454(18)30068-5
DOI:
10.1016/j.actamat.2018.01.037
Reference:
AM 14331
To appear in:
Acta Materialia
Received Date: 10 November 2017 Revised Date:
29 December 2017
Accepted Date: 2 January 2018
Please cite this article as: X. Zhou, J. Song, Effects of alloying elements on vacancies and vacancyhydrogen clusters at coherent twin boundaries in nickel alloys, Acta Materialia (2018), doi: 10.1016/ j.actamat.2018.01.037. 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 proof before it is published in its final 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.
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Effects of Alloying Elements on Vacancies and Vacancy-Hydrogen
Xiao Zhou1, Jun Song1,* 1.
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Clusters at Coherent Twin Boundaries in Nickel Alloys
Department of Mining and Materials Engineering,
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Abstract
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McGill University, Montréal, Québec, H3A 0C5, Canada
The interactions between several typical alloying elements (i.e., Al, Cr, Mo, Nb and Ti) and hydrogen at coherent twin boundaries (CTB) in Ni alloys were systematically studied through first-principles calculations. It was found that solute atoms generally prefer to segregate at CTBs, with the segregation tendency prescribed by the size
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mismatch between the solute and host Ni atom. Moreover, there exist attractive interaction between certain solute atoms, potentially promoting solute co-segregation at CTBs. Though the solute presence does not favor hydrogen accumulation, it can considerably reduce the formation energies of vacancies and vacancy-hydrogen clusters
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at CTBs, which, if augmented by large-scale plastic deformation, may facilitate nanovoid
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nucleation to promote eventual crack initiation at CTBs. The present study demonstrated the critical role of alloying in hydrogen-induced crack initiation at CTBs, providing a new perspective towards understanding HE in Ni alloys.
Keywords: Hydrogen emrbittlement; coherent twin boundary; solute segregation; vacancy-hydrogen cluster; nanovoid nucleation. *
Corresponding author. E-mail address: [email protected] (Jun Song)
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1. Introduction Hydrogen-induced degradation of materials has been a subject of continuous interest
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for many decades [1-3]. In particular, the degradation by hydrogen embrittlement (HE) [4-6], which causes many high-strength metals, such as high strength steels and Ni alloys,
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to undergo premature failure with a drastic loss in ductility, toughness and strength [2, 7, 8], has been a long-standing industrial nuisance. Albeit enormous research efforts, there is
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still no general theory, nor consensus on the mechanisms underlying HE [5, 9-13]. HE in polycrystalline metals is further complicated, and often manifested, by the presence of microstructures, such as grain boundaries (GBs), which affect hydrogen diffusion and segregation, and modify the local lattice cohesion and deformation behaviors[14-16].
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Bechtle et al. [17] demonstrated the paramount importance of GBs in controlling the material’s susceptibility to HE.
They showed that the severity of HE of pure Ni can be
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significantly reduced by engineering the microstructure to enhance the fraction of special GBs that consist of principally Σ3 twin boundaries. Among the various special GBs, the
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coherent twin boundary (CTB) is of particular interest. With nearly identical atom arrangement as the bulk lattice, CTBs are often regarded to be highly resistant to fracture and HE [18, 19]. Nonetheless, contradicting results regarding the role of CTBs exist in the literature. In the study by Ulmer and Altstetter [20] on hydrogen-assisted fracture in austenitic stainless steels, it was found that cracking frequencies at annealing twins and other GBs were approximately the same. Seita et al. [21, 22] performed in situ tensile 2
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tests on hydrogen charged Ni-based superalloy samples and identified CTBs as the preferential microstructural sites for crack initiation.
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The above conflicting opinions about CTBs are intriguing and suggest the necessity of further studies. One puzzle piece absent from those previous studies is the effect of alloying elements (thereafter also referred to as solute atoms below). With alloying being
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commonly used to modulate the mechanical properties and corrosion resistance of Ni
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alloys, solutes are unavoidably present in the Ni lattice [23-25]. As reported by many previous studies [26-28], solute atoms may segregate at GBs. Therefore understanding of HE at CTBs necessitates the investigation of solute-hydrogen interplay at CTBs. In this regard, here we study the role of typical solute atoms (i.e., Al, Cr, Mo, Nb and
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Ti) in affecting HE at CTBs in Ni alloys, employing first-principles calculations. Their segregation and co-segregation at CTBs, and interactions with hydrogen were examined.
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Meanwhile, a few recent studies on Ni-based alloys [29, 30] suggest that microvoids serve as nucleation sites for crack initiation sites along GBs in hydrogen-charged samples.
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Consequently we investigated the coupled effect of solute and hydrogen on vacancy accumulation and subsequently nanovoid nucleation at CTBs. In the end, the implication of solute presence to hydrogen-induced CTB cracking and HE susceptibility of Ni alloys was discussed.
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2. Computational Methodology First-principles calculations on the basis of density functional theory (DFT) using the
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Vienna ab-initio simulation package (VASP) [31, 32] were performed to examine the interplay between alloying elements with vacancies and vacancy-hydrogen clusters.
atoms.
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Here a vacancy-hydrogen cluster refers to a vacancy bound with one or several hydrogen For simplicity, below we denote the vacancy-hydrogen cluster as VHx with x In the calculations, the electron-ion
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being the number of hydrogen atoms in the cluster.
interactions were described by Blöch projector-augmented wave method (PAW) [33, 34], and Generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [35] based on plane-wave basis sets was employed for electron exchange and correlation For calculations involving a bulk lattice, a 3×3×3 periodic supercell
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functionals.
containing 108 atoms was used, with a k-point grid of 5×5×5 and the plane-wave cut-off
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energy being 400 eV, while for calculations involving CTB, a periodic supercell model of dimensions 8.8Å×5.0Å×24.4Å containing 96 atoms, constructed based on coincidence
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site lattice theory (CSL) [36] was used with a k-point grid of 5×5×10 and a cutoff energy of 380 eV.
In all calculations, the convergence criteria of energy and atomic force were
set as 10-5 eV and 10-2 eV/Å respectively, and the first-order Methfessel-Paxton scheme with a 0.1 eV smearing width was adopted. The supercell of CTB used in this study is schematically illustrated in Fig. 1.
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The alloying elements were introduced into the Ni lattice by substituting host Ni atoms. The alloy elements considered in our study are Al, Cr, Mo, Nb and Ti, which are
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widely used in commercial Ni-based alloys (e.g., Inconel 725 [21]) for improving the mechanical properties. The solution energy of an isolated alloying element, denoted as ES , is defined as
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ES = Etot [ Nin -1S ] − Etot [ Nin ] − ES0 + E Ni0 ,
(1)
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where Etot [Ni n ] is the total energy of the reference bulk or CTB Ni system, Etot [Ni n −1S ] is the total energy of the corresponding bulk or CTB Ni system where one Ni atom is substituted by an alloying element, ES0 is the cohesive energy of the alloying atom in its stable crystalline state and ENi0 is the cohesive energy of Ni. Apparently the solution
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energies of a solute in the bulk and at CTB may differ.
The difference can be
characterized the segregation energy, ESseg , defined as (2)
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ESseg = ESCTB − ESBulk
where ESCTB and ESBulk denote the solution energies of the solute located at CTB and in
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the bulk respectively. ESseg effectively indicates the tendency of the solute to segregate at CTB.
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Fig. 1 (Color online): Schematic illustration of the CTB model used in DFT calculations in this study. The green and grey spheres indicated atoms at the CTB and in the bulk respectively. The circle with dash line represented the vacancy and the small red balls indicated the positions of hydrogen atom. The left and right figures above are projection views the system along [110] and [111] directions respectively.
The interaction between hydrogen and a solute atom S, EbSH , is defined as EbSH = E [ Ni n −1 SH ] − E [ Ni n −1 S ] − E [ Ni n H ] + E [ Ni n ] ,
(3a)
where E [ Ni n −1SH ] is the total energy of a (bulk or CTB) Ni lattice containing n-1 Ni
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atoms, one solute atom S and one hydrogen, E [ Ni n −1S ] and E [ Ni n H ] are the total energies of a lattice containing n-1 Ni atoms plus one solute atom S and a lattice that
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holds n Ni atoms plus one hydrogen atom respectively, and E [ Ni n ] is the total energy
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of the reference Ni lattice. Eq. 3a can also be generalized for the situation of multiple hydrogen adsorption, as the following: EbSH = E [ Ni n−1 SH m ] − E [ Ni n−1 SH m −1 ] − E [ Ni n H ] + E [ Ni n ] ,
(3b)
Meanwhile the interaction between solute atoms can be analyzed in a similar way, with the binding energy Ebαβ between two solute atoms, α and β , in the Ni lattice defined as:
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Ebαβ = E [ Ni n − 2αβ ] − E [ Ni n −1α ] − E [ Ni n −1 β ] + E [ Ni n ] ,
(4)
where E [ Ni n − 2αβ ] is the total energy of a (bulk or CTB) Ni lattice containing n-2 Ni
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atoms and two solute atoms α and β , and E [ Ni n −1 x ] ( x = α or β ) is the total energy of a Ni lattice containing n-1 Ni atoms and one solute atom x.
In our definitions (i.e.,
Eqs 3-4 above), positive and negative values of the binding energy indicates repulsion
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and attraction respectively between hydrogen and the solute atom, or two solute atoms.
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The vacancy was created by removing a host Ni while a vacancy hydrogen cluster VHx is constructed by introducing hydrogen atoms into interstitial sites neighboring the vacancy. For a vacancy (essentially VHx with x = 0) or VHx, the formation energy E VH f is defined as VH x
= E [ Ni n −1 , VH x ] + ENi0 − E [ Ni n ] −
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Ef
x
2
EH0 ,
x
(5)
2
where E [ Ni n −1 , VH x ] is the total energy of a system containing n-1 host Ni atoms and 1
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VHx cluster, E [ Ni n ] is the total energy of the corresponding Ni lattice that contains n host Ni atoms. The above formation energy of VHx, in the presence of solute atoms, is
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modified and becomes
E f , sol = E [ Ni n − m −1S m , VH x ] + ENi0 − E [ Ni n − m S m ] − VH x
x 2
EH0 ,
(6)
2
where E [ Ni n − m −1S m , VH x ] is the total energy of the system containing n-m-1 host Ni atoms, m solute atoms, and one VHx cluster, E [ Ni n − m S m ] is the total energy of the system containing n-m host Ni atoms and m solute atoms. The difference between E VH and f x
E VH thus provides a metric to assess the influence of solute on the formation of VHx: f , sol x
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VH x x x ∆E VH = E VH . f f , sol − E f
(7)
Note that for the VHx cluster, only hydrogen segregation at the nearest-neighboring Meanwhile in
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octahedral sites surrounding the vacancy was considered, and thus x≤6.
our investigation of solute-vacancy and solute-VHx interactions, we only consider the situation of solute atoms residing in the immediate vicinity (i.e., at nearest neighboring In addition, one thing worth noting in the above
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sites) of the vacancy or VHx.
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calculations is that there can be multiple configurations for a particular solute-hydrogen, solute-solute, solute-vacancy or solute-VHx interaction, and we only considered the configuration of the lowest energy.
Meanwhile we also examined the segregation tendency of VHx towards CTB.
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x (CTB) and Denoting the formation energies of VHx at CTB and in bulk Ni as E VH f
x (Bulk) E VH respectively, we can define the segregation energy of VHx, in a similar way as f
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that of the segregation of solute atom:
VH Eseg = E VH f x
x
( CTB)
− E VH f
x
( Bulk)
(8)
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In the case where CTB has pre-segregated solute atoms, the segregation energy, now VH x denoted as Eseg , sol , becomes VH VH ( CTB) Eseg − E VH ,sol = E f ,sol f x
x
x
( Bulk)
,
(9)
( CTB) represents the formation energy of VHx at CTB, affected by the where E VH f ,sol x
VH x x ( CTB) presence of solute. One fact to note is that E VH and Eseg are related to each f , sol
other via ∆E VH (at CTB), a term previously defined in Eq. 7, which can be expressed as f x
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VH x VH x VH x Eseg , sol − Eseg = ∆E f
(10)
3.1 Solute atoms and VHx in bulk Ni
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3. Results and discussion
We first examined the energetics of the solute atom, VHx and their mutual
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interactions in the bulk Ni lattice, in order to establish a baseline for our subsequent study
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of the CTB.
3.1.1 Solute-solute and solute-hydrogen interactions in bulk Ni
Considering two nearest-neighboring solute atoms, we calculated the corresponding binding energies, shown in Fig. 2. Almost exclusively for all solute pairs (except for the
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case of Cr-Cr which shows a tiny negative Eb of -0.03 eV), the binding energy exhibits a positive value, indicating that solute atoms do not want to be in the immediate neighbors
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of each other, and thus no tendency for solute atoms to aggregate into clusters in the bulk Ni lattice. The binding energies of solute-hydrogen interactions (with hydrogen located
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at an interstitial site immediately neighboring the solute atom) are shown in Fig. 2b, again showing positive values, suggesting a repulsive interaction between solute and hydrogen atoms.
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Fig. 2: Binding energies of different (a) solute-solute ( , see Eq. 3a) couples in bulk Ni.
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(
, see Eq. 4) and (b) solute-hydrogen
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3.1.2 Effect of solute atoms on the formation of VHx in bulk Ni
The formation energies of VHx in a solute-free bulk Ni lattice are listed in Table I.
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The results are in good agreement with previous studies [37-41]. Fig. 3 illustrates the effect of solute atoms (i.e., ∆E VH , see Eq. 7) on the formation of VHx. Note that here f x
we only considered the presence of individual solute atoms in the immediate vicinity of a VHx cluster, because of the repulsion between solute atoms (cf. Fig. 2a). In addition, it is worth noting that only those nearest octahedral interstitial sites neighboring the vacancy
10
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are considered for potential hydrogen trapping and consequently the maximum number of
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trapped hydrogen is six.
Table I: The formation energies of vacancy (i.e., VHx with x=0) and vacancy hydrogen clusters (i.e., VHx with x=1-6) in pure Ni, in bulk lattice (
Data from literature 1.48 , 1.40[38], 1.14[39], 1.67[40], 1.44[41] 1.35[37], 1.21[38] 1.21[37], 1.01[38] 1.07[37], 0.87[38] 0.93[37], 0.70[38] 0.79[37], 0.57[38] 0.65[37], 0.33[38] [37]
(
)
).
x (CTB) E VH (eV) f
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This work 1.40 1.26 1.10 0.98 0.85 0.72 0.57
) and at CTB (
This work 1.42 1.28 1.05 0.94 0.80 0.67 0.53
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0 1 2 3 4 5 6
x (Bulk) E VH (eV) f
)
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Number of hydrogen
(
As can be seen in Fig. 3, for x between 0 and 4, the formation of VHx is favored by <0) while impeded by the existence of Cr and Mo the presence of Al, Nb or Ti ( ∆E VH f
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x
( ∆E VH >0). In the cases of x=5 and 6, however, all solute atoms show positive ∆E VH f f x
x
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values, resisting the formation of VHx.
11
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Fig. 3 (Color online): The effect of solute atoms, reflected by ∆
(see Eq. 7), on the
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formation of vacancy hydrogen cluster VHx as x (i.e., Number of H) varies.
3.2 Solute atoms and VHx at CTB
3.2.1 Segregation of solute atoms at CTB
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For the case of CTB, we first examined the segregation tendency of solute atoms. As seen in Fig. 4, all solute atoms exhibit negative ESseg , indicative of it being energetically preferable for solutes to segregate at CTB. Fig. 4 also indicates that those solute atoms
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have different segregation tendency, being strongest ( ESseg = -0.10 eV) for Nb and
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weakest for Cr ( ESseg = -0.05 eV).
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Fig. 4 (Color online): (Top) DFT calculated and continuum predicted (see Eq. 13) segregation energies of different solute atoms at CTB. (Bottom) The elastic energy and size factor Φ corresponding to different solute atoms. The lines are drawn to guide the eye.
The difference can be attributed to that the energetics of substitutional solute atoms are greatly influenced by the size mismatch between the solute and host atoms, as well
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acknowledged by many previous studies [42-44]. Denoting V0 and Vs as the atomic
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volume of a host Ni atom and solute atom respectively, the size mismatch can be quantified via the size factor ( Φ ): Φ=
Vs − V0 V0
.
(11)
Note that the V0 of a host Ni atom can be approximated by its Voronoi volume [45], being different for Ni atoms in the bulk and at CTB. The size mismatch leads to lattice distortion and subsequently gives rise to an elastic energy Eel, which can be analyzed 13
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within the framework of continuum mechanics as proposed by Eshelby [42, 46], treating the solute substitution process as the insertion of a spherical inclusion of volume Vs into a
Eel =
Bs ( ∆Vs ) 2 2Vs
+
2G0 (∆V0 ) 2 3V0
.
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hole of volume V0 in the host matrix : (12)
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where ∆Vs and ∆V0 are the volume changes of spherical inclusion and hole due to the internal stress, Bs is the bulk modulus of inclusion, and G0 is the shear modulus of host
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matrix. Considering the stress continuity at the inclusion-matrix interface, the above can be rewritten as [42]:
Eel =
2 Bs G0 (Vs − V0 )2 . 3BsV0 + 4G0Vs
(13)
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With V0 being different for the bulk and CTB, Eel thus exhibits different values (denoted as E elB and EelCTB below) for a substitutional solute in the bulk and CTB, and
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el the difference, Eseg (see Eq. 14 below), effectively defines the segregation energy
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predicted from continuum mechanics: el Eseg = EelCTB − EelB .
(14)
The values of size factor Φ , elastic energy EelCTB and continuum-predicted segregation el el energy Eseg of solute atoms at CTB are all presented in Fig. 4. We note that Eseg
exhibits essentially the same trend as the DFT-calculated segregation energy ESseg , albeit some difference in the absolute value. This observation suggests that the mechanical
14
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energy be the dominant factor in determining the segregation tendency of solute atoms towards CTB.
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The preferential segregation will certainly lead to enrichment of solute atoms at CTB. To have an idea of the level of solute enrichment at CTB, we consider two Ni alloys, Inconel 718 and 725, as representatives, and the concentration ( CSCTB ) of each alloy
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element at CTB is roughly estimated based on the following equation:
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CSCTB
− E seg CSBulk exp S k BT , = − ESseg Bulk 1 + CS exp k BT
(15)
where C SBulk is solute concentration in the bulk lattice and kB is Boltzmann constant and
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T is temperature. CSBulk is taken as either the alloy element solubility (obtained from FactSage alloying database [47, 48]) or alloy composition [22, 49], whichever is lower. Using the segregation energies obtained above (see Fig. 4), the estimated atomic
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concentrations of solute atoms (at room temperature) expected at the CTB are listed in
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Table II, from which we can see that in equilibrium appreciable amount of Cr, Ti, Mo and Nb would be present at CTB. It is worth to note that the above calculation is a rough estimation without accounting for the mutual interaction and competition between different solute elements. As a result, the CSCTB value obtained from Eq. 15 serves more of an indication of the solute segregation tendency, and can differ from the actual solute
15
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concentration at CTBs in those Ni alloys (which would depends on the complex interplay between solutes and the processing conditions).
Element
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Table II: Bulk solute concentration and corresponding solute segregation expected at CTB for two representative Ni alloys, Inconel 718 and 725. Inconel 718
Inconel 725
Atomic concentration (at. %)
Atomic concentration (at. %)
CTB
Bulk†
Cr
4.24
22.69
4.24
Ti
1.35
14.58
Nb
2.76
54.44
Mo
0.51
13.39
Al
0.76
7.12
CTB
22.69
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Bulk
1.97
19.94
2.41
51.07
0.51
13.39
0.65
6.16
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†
† Note that the bulk concentration is approximately taken as either the alloy solubility (at 300K) or alloy composition, whichever is lower.
3.2.2 Solute-solute and solute-H interactions at CTB
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The appreciable segregation of solute atoms would unavoidably lead to mutual interaction between solute atoms. Fig. 5a shows the binding energies of different solute
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pairs. Unlike the case in bulk Ni (cf. Fig. 2), Ebαβ exhibits negative values for several pairs (with EbMoNb = -0.25 eV being the lowest, see Fig. 5a), indicating the possibility of
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solute co-segregation at CTB.
The interaction between hydrogen and individual solute atoms at CTB is
demonstrated in Fig. 5b. Here in our calculation (up to six) hydrogen atoms were introduced to the interstitial sites immediately neighboring a solute atom at CTB, and the most energetically stable configurations were determined, from which the binding energy
EbSH was then obtained. As seen in Fig. 5b, EbSH generally exhibits positive values, 16
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indicative of repulsion between hydrogen and individual solute atoms. The repulsion becomes stronger as the number of H increases. Meanwhile, the interaction between
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hydrogen and solute pairs was also examined and EbSH was also found to remain positive, as shown in Fig. 6. The above results demonstrate that solute segregation alone
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would impede hydrogen accumulation at CTB.
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Fig. 5 (Color online): (a) Binding energies ( Binding energies (
, see Eq. 4) of different solute pairs at CTB. (b)
, see Eqs 3a-b) between hydrogen and individual solute atoms at CTB.
17
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Fig. 6: Binding energies ( at CTB.
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, see Eq. 3a) between one hydrogen atom and different solute pairs
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3.2.3 Effect of solute atoms on the formation of VHx at CTB
Fig. 7: The effect of (a) individual solute atoms and (b) solute pairs, reflected by ∆
(see Eq.
7), on the formation energy of a single vacancy (i.e., VHx with x = 0) at CTB.
18
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The formation energies of VHx at CTB in pure Ni are listed in Table I. As seen in the table, they show negligible difference in comparison to those in pure bulk Ni. Below
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we elaborate on how the formation of VHx at CTB is affected by the presence of solute. For a single vacancy (i.e., VH0) at CTB, the influence of individual solute atoms and solute pairs on its formation is shown in Fig. 7. Note for solute pairs we only consider
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those with tendency of co-segregation (i.e., Ebαβ being negative, see Fig. 6 above) at
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CTB. For the case of individual solutes, we note from Fig. 7a that while Mo and Cr increase the vacancy formation energy, Al, Ti and Nb reduce the vacancy formation energy with Nb leading to the largest reduction of 0.15 eV. For the case of solute pairs, the Cr-Mo increases the vacancy formation but other pairs considerably lower the
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vacancy formation energy. In particular, we see (cf. Fig. 7b) that the Nb-Ti pair has the most significant effect, reducing the vacancy formation energy by 0.26 eV (~18%
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reduction). Overall the presence of solutes and solute pairs predominantly tends to reduce
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the vacancy formation energy, facilitating vacancy formation at CTB.
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∆
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Fig. 8 (Color online): The effect of (a) individual solute atoms and (b) solute pairs, reflected by (see Eq. 7), on the formation energies of vacancy hydrogen clusters VHx at CTB.
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With hydrogen present, the effect of solute and solute pairs on the formation of VHx is illustrated in Fig. 8. For individual solutes (cf. Fig. 8a), we note that Nb and Ti, and Al
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can effectively reduce the formation energy of VHx with x up to 4 and 2 respectively. Meanwhile Cr can also reduce the formation energy of VHx with x ≤ 3, with the effect being much less pronounced. In the contrary, the presence of Mo always inhibits VHx x formation. For VHx with x ≥ 5, all solute atoms lead to positive ∆E VH , resisting the f
formation of VHx.
For solute pairs (cf. Fig. 8b), we see that Mo-Nb (slightly), Al-Nb
and Nb-Ti favor the formation of VHx with x = 1, x ≤ 2 and x ≤ 3 respectively, while 20
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otherwise the formation of VHx is impeded by the presence of solute pairs.
Overall,
from Fig. 8, we see that the presence of solute (individual or pair) can greatly inhibit
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excessive hydrogen aggregation at vacancies but may promote the formation of “light” vacancy-hydrogen bundles, in particular VH1 and VH2.
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3.6 Synergetic effect of hydrogen and solute under plastic deformation
In previous sections, we have demonstrated that the formation of vacancies and VHx
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clusters can be notably affected by the presence of solutes, and in certain cases greatly facilitated. Normally the vacancy concentration at ambient temperature is very low in annealed structural metal samples (e.g., ~10-23 in Ni [37]), in which case hydrogen and
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solutes would play limited role in modifying the mechanical behaviors of metals. However, excess vacancies can be induced by plastic deformation. For instance, Zehetbauer et al. [50, 51] demonstrated that the concentration of vacancies produced in
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Ni and Cu by severe plastic deformation at room temperature can reach the order of 10-4
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or higher, close to the thermal equilibrium concentration of vacancies at the melting temperature. As suggested by Militzer et al. [52], the net rate of generation of plasticity-induced excess vacancies can be expressed as:
dCV = Γ −ξ , dt
(16)
21
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where CV denotes the concentration of excess vacancies, and Γ and ξ are the production and annihilation rate respectively. The production rate Γ can be expressed
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as [52]: (17)
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where χ is a dimensionless constant, being approximately 0.1 [53, 54], Ω 0 is the
Q f is the vacancy formation energy.
is the strain rate and
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atomic volume (being 1.09×10-29 m3 for Ni), σ is the stress,
In the presence of hydrogen atoms, those plasticity-induced vacancies may be stabilized to remain in the lattice during the deformation process. As suggested by Li et al.
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[13], under the condition that the concentration of hydrogen exceeding vacancy population CV, most of the vacancies will combine with hydrogen atoms to form VHx clusters, which prevents them from easy annihilation and effectively renders ξ zero as a
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first-order approximation, consequently leading to
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(18)
Note that in this equation CV includes both vacancies and VHx clusters. By integrating the above equation, one linear relation between CV and ε can be obtained:
CV =
χΩ 0σ Qf
ε + C0 ,
(19)
22
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where C0 is the thermal equilibrium vacancy concentration in Ni bulk lattice expected in absence of plastic deformation (~10-23 at room temperature [37]).
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From Eq. 19, the bulk concentration of plasticity induced excess vacancies CV in presence of hydrogen can be determined as a function of σ and ε .
As an example,
Fig. 9 shows the mapping of CV (being the ratio of vacancies over Ni atoms) for Inconel
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725 for the ranges of stress/strain of typical stress-strain curves under hydrogen charging
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at room temperature [21].
Fig. 9 (Color online): The plastic deformation induced vacancy concentration in bulk Ni lattice, Cv according to Eq. 17, within the typical stress-strain range for Inconel 725 [21].
As seen from Fig. 9, the plastic deformation result in CV to range from 10-4 to 10-3, being significantly higher than the normal vacancy concentration (~10-23) expected at
23
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room temperature in absence of plasticity. This substantial bulk presence of vacancies will be further manifested at CTB and affected by solute atoms, as suggested by our
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previous results. Denoting the concentration of vacancies (including VHx) at CTB as CVCTB , it can be roughly estimated as
(20)
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CVCTB
VH x − Eseg , sol CV exp k T B CTB + C CTB , = CS × V0 VH x − Eseg , sol 1 + CV exp k T B
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VH x where CSCTB denotes the solute or solute pair concentration at CTB, Eseg , sol (cf. Eqs.
9-10) denote the segregation energy of vacancy-hydrogen cluster at CTB in the presence of solute. Note here for simplicity we assume pre-segregation of solute atoms at CTB
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before the segregation of vacancy-hydrogen clusters occurs. The term CVCTB denotes the 0 total vacancy concentration expected at CTB in absence of solute, related to CV as:
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CVCTB 0
− E VH x CV exp seg kT B , = CV × VH x − Eseg 1 + CV exp k T B
(21)
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VH x with Eseg previously defined in Eq. 8.
As previously elaborated (in Section 3.2.3), certain solute atoms (i.e., Cr, Ti, Nb and
Al) or solute pairs (i.e., Nb-Ti, Al-Nb, Al-Ti and Cr-Nb) can effectively reduce the formation energy of single vacancies and/or VHx (for some x), therefore possibly contributing to enhancing the total vacancy population. Fig. 10 and Fig. 11 shows the calculated CVCTB under different individual solute (i.e. C SCTB ) and solute pair 24
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concentration levels, as a function of CV. We note that the total vacancy population at CTB can be substantially elevated by the solute presence, being particularly significant in
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the presence of Nb, Al, Nb-Ti and Al-Nb. These results suggest that under plastic deformation, the synergetic effect of hydrogen and solute may promote considerable
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vacancy aggregation at CTBs.
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Fig. 10 (Color online): Total vacancy population at CTB ( ), as a function of bulk vacancy concentration (CV), influenced by the presence of different levels of individual solutes at CTB.
25
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Fig. 11 (Color online): Total vacancy population at CTB ( ), as a function of bulk vacancy concentration (CV), influenced by the presence of different levels of solute pairs at CTB.
One consequence that naturally follows the enhanced vacancy concentration at CTB is the increasing possibility of vacancy coalescence, which would subsequently lead to
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nanovoid nucleation. Previous studies have well demonstrated the critical relation
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between vacancy concentration and (nano)void nucleation [30, 55]. Reina et al. [56] formulated a model of nanovoid nucleation under severe plastic deformation, relating the nucleation to diffusion-mediated vacancy aggregation and subsequent vacancy coalescence. In their study, it was postulated that a certain time tf is required to establish a steady-state concentration CV of vacancies:
tf =
CV
v0 exp − ( EVf + E if ) / kBT
,
(22)
26
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where v0 is the Debye frequency, and
and
!
denote the formation energies of a
vacancy and far-field self-interstitial respectively. The time tf can be regarded as a lower
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bound of the time required for the nanovoid nucleation. Below we perform a simple analysis to assess the effect of solute on tf, thus roughly gauging its influence on the kinetics of nanovoid nucleation.
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Applying the above equation to CTB, we can obtain the required times to achieve a threshold concentration of solute atoms, respectively as: t
=
CTB f
CVCTB
,
(23a)
CVCTB
,
(23b)
v0 exp − ( EVf + E if ) / k BT
v0 exp − ( E Vf , s + E if ) / k BT
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t CTB f ,s =
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of vacancies at CTB, without and with the presence of
from which we may define a normalized time parameter as:
t CTB f
= exp ∆E f / k BT ,
(24)
is essentially ∆E VH defined previously in Eq. 7. We see that tn is dependent f x
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where Δ
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tn =
t CTB f ,s
on the solute-induced change in the vacancy formation energy. As illustrated in Fig. 12, certain solutes or solute pairs can render tn << 1 for vacancy and vacancy-hydrogen clusters. This indicates that the presence of solute atoms may contribute to decreasing the equilibration time required for attaining a particular stead-state vacancy concentration at CTBs, thereby facilitating nanovoid nucleation. The accelerated nanovoid nucleation,
27
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together with possible localization of dislocation activities at CTBs [57], eventually may assist or promote crack initiation. Furthermore, the above scenario would also be
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generally applicable to other GBs.
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Fig. 12 (Color online) The normalized time tn (see Eq. 24, at T=300K) corresponding to single vacancy (VH0) and vacancy-hydrogen clusters (VH1 and VH2), influenced by the presence of individual solutes (top) and solute pairs (bottom) at CTBs. The lines are drawn to guide the eye.
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4. Conclusions
In summary, comprehensive first-principles calculations have been performed to
investigate the interplay between hydrogen and typical alloying elements (i.e., Al, Cr, Mo, Nb and Ti) in Ni alloys, and their coupled effect on the formation of vacancy and vacancy-hydrogen clusters at coherent twin boundaries (CTBs) in Ni alloys. The key findings are listed as the follows:
28
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1) It is energetically preferable for solute atoms to segregate at CTBs, with the segregation tendency directly prescribed by the size mismatch between the solute
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and host Ni atom; 2) Unlike the case of bulk Ni lattice where different solute atoms repel each other, at CTBs there exists attractive interaction between certain solute atoms, potentially
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promoting co-segregation;
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3) The solute presence at CTB generally inhibits hydrogen accumulation, but can considerably reduce the formation energies of vacancies and vacancy-hydrogen clusters;
4) This considerable reduction in the formation energy, when augmented by large-scale
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plastic deformation, may lead to marked increase of total vacancy population at CTBs, and subsequently facilitate nanovoid nucleation to aid eventual crack
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initiation at CTBs.
The present study elucidates possible implication of alloying to hydrogen-induced CTB
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cracking, providing a new perspective towards understanding HE in Ni alloys.
Acknowledgements XZ and JS would like to acknowledge financial support by Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant # RGPIN
29
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418469-2012), National Natural Science Foundation of China (NSFC Grant No. 51628101) and China Scholarship Council (CSC), and thank Supercomputer Consortium
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Laval UQAM McGill and Eastern Quebec (CLUMEQ) for providing computing resources.
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