On the nucleation of planar faults during low temperature and high stress creep of single crystal Ni-base superalloys

Accepted Manuscript On the nucleation of planar faults during low temperature and high stress creep of single crystal Ni-base superalloys X. Wu, A. Dlouhy, Y.M. Eggeler, E. Spiecker, A. Kostka, C. Somsen, G. Eggeler PII:

S1359-6454(17)30833-9

DOI:

10.1016/j.actamat.2017.09.063

Reference:

AM 14093

To appear in:

Acta Materialia

Received Date: 29 June 2017 Revised Date:

25 September 2017

Accepted Date: 27 September 2017

Please cite this article as: X. Wu, A. Dlouhy, Y.M. Eggeler, E. Spiecker, A. Kostka, C. Somsen, G. Eggeler, On the nucleation of planar faults during low temperature and high stress creep of single crystal Ni-base superalloys, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.09.063. 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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Graphical abstract

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Nucleation of stacking fault in single crystal Ni-base superalloy

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On the Nucleation of Planar Faults during Low Temperature and High Stress Creep of Single Crystal Ni-base Superalloys

X. Wu1*, A. Dlouhy2, Y.M. Eggeler3, E. Spiecker3,

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A. Kostka4, C. Somsen1, G. Eggeler1

Revised Manuscript, 23. September 2017

Institut für Werkstoffe, Ruhr-Universität Bochum, 44780 Bochum, Germany

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Institute of Physics of Materials, ASCR, Žižkova 22, 616 62 Brno, Czech Republic

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Department Werkstoffwissenschaft, Friedrich-Alexander-Universität, 91058 Erlangen, Germany

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Zentrum für Grenzflächendominierte Höchstleistungswerkstoffe, ZGH, RuhrUniversität Bochum, 44780 Bochum, Germany

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* corresponding author: [email protected]

Abstract

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The present work studies the nucleation of planar faults in the early stages of low temperature (750°C) and high stress (800 MPa) creep of a Ni-base single crystal

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superalloy (SX). Two families of 60° dislocations w ith different Burgers vectors were detected in the transmission electron microscope (TEM). These can react and form a planar fault in the γ’ phase. A 2D discrete dislocation model helps to rationalize a sequence of events which lead to the nucleation of a planar fault. First, one 60° channel dislocation approaches another 60° interfac e dislocation with a different Burgers vector. At a distance of 5 nm, it splits up into two Shockley partials. The interface dislocation is pushed into the γ’-phase where it creates a small antiphase boundary. It can only move on when the leading Shockley partial joins it and creates an overall 1/3<112> superdislocation. This process is fast and therefore is difficult to observe. The results obtained in the present work contribute to a better

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ACCEPTED MANUSCRIPT understanding of the processes which govern the early stages of low temperature and high stress primary creep of SX.

Keywords: Ni-base superalloy single crystals, low temperature and high stress

Introduction

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primary creep, nucleation of planar faults, gamma channel dislocations

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There has been ongoing keen interest in the elementary processes which govern creep of Ni-base single crystal superalloys (SX). SX are used to make turbine blades

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which have to withstand mechanical loads at extremely high temperatures [e.g. 1-8]. Their microstructure consists of cuboidal γ’-particles which are separated by thin γchannels. In the early stages of creep, dislocation activity is confined to the γchannels [e.g. 9-16]. Later, dislocations cut into the γ’-phase [14,17]. Transmission electron microscopy (TEM) studies of the evolution of dislocation substructures in single phase Ni3Al [e.g. 18-21] do not help to explain dislocation reactions in the γ’-

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phase of SX, because they do not account for the local internal stresses associated with the crystallographic misfit between the γ- and γ’-phase [13,23,24]. In the past, dislocation processes which govern SX creep in the high temperature (T) low stress (σ) creep regime have been thoroughly investigated [15,16,25-29] and there are no

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controversial views about the elementary dislocation processes which govern creep. Dislocation processes in the low T and high σ regime have been reported to be more

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complex [24,30-57]. In this creep regime the importance of <112>{111} shear is well appreciated. Dislocation ribbons with an overall a<112> displacement vector propagate through the γ/γ’-microstructure [30,44,46]. Four 1/2<110> γ-channel dislocations on a common {111} plane must react to form such a ribbon [e.g. 42,44,46].

Recently the low temperature high stress phenomenon of double minimum creep was studied [14]. Coupled elementary dislocation mechanisms were proposed which rationalize a small initial period of decreasing creep rate in the very early stages of creep. In this period γ-channel dislocation plasticity has established optimum conditions for γ’-phase stacking fault cutting [44,46,47]. In their seminal publications, 2

ACCEPTED MANUSCRIPT Kear et al. [e.g. 30-33] propose a scenario, where three 1/6<112> type partial dislocations in the γ-phase approach the γ/γ’-interface. The two leading partials jointly cross the interface and form a 1/3<112> γ’-superpartial, which creates a SISF in its wake. The third trailing partial remains in the γ-channel at the γ/γ’-interface. When a second triple of the same type of γ-channel partial dislocations approaches, the

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trailing partial of the first triple and the leading partial of the second triple enter the γ’phase. These two super partial dislocations are APB-coupled. Once the leading partial of the second triple enters the γ’-phase, it creates a superlattice extrinsic stacking fault (SESF). When the trailing two partials of the second triple cross the

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γ/γ’-interface, they combine and form a 1/3<112> superpartial, which removes the SESF and restores order. We refer to this sequence of elementary events as Kear-

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scenario.

In the mid-eighties, the field of low temperature high stress γ’-cutting in SX was revisited [35-39]. Weak beam TEM results did not support the Kear-scenario. Most importantly, 1/2<112> dislocations were not observed in the γ-channels where only 1/2<110> dislocations were present. Décamps and co-workers suggested a scenario, where a single 1/2<110> γ-dislocation cuts into the γ’-phase and creates an

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antiphase boundary (APB) [35,37,39]. Subsequently, a 1/6<112> Shockley partial dislocation loop nucleates. One part of this dislocation loop reacts with the leading 1/2<110> dislocation and forms a 1/3<112> super partial dislocation. The other part

SISF.

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moves back to the γ/γ’-interface and converts the high energy APB into a low energy

Sass et al. [40,41] observed two 1/2<110> type dislocations with dissimilar Burgers

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vectors on one common glide plane which jointly cross the γ/γ’-interface. Then they combine and create a leading 1/3<112> superpartial dislocation, a SISF and a trailing 1/6<112> superpartial dislocation. This cutting required the cross slip of one screw dislocation in the γ-channel. The probability of such a reaction to occur between two 60°-dislocations (which govern <100> tensile creep) was considered to be low [40,41]. Rae et al. [44,46,47] confirmed the deformation of superalloys by <112> ribbons for CMSX-4. They pointed out that when a <112> dislocation ribbon moves through the γ/γ’-microstructure one must understand the dislocation processes in both phases, especially the role of γ-channel dislocation activity which precedes γ’phase cutting. 3

ACCEPTED MANUSCRIPT Chen and Knowles [45] suggested that for specific SX orientations, SESFs in larger γ’-particles coexist with micro twins. They proposed that the nucleation of both, SESFs and micro twins, is associated with the interaction of two different 1/3<112> partials on every {111} plane. They claim that shear loading in the opposite direction would create SISFs and that micro twins can only be observed in shear directions

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where SESFs form. Kovarik et al. [48,49] confirm an early suggestion by Kolbe [50]. They showed that the operative twinning dislocations are identical Shockley partials of type 1/6<112>, which propagate through the γ’-phase in closely-separated pairs on consecutive {111} planes. The rate-limiting elementary process is diffusion controlled

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re-ordering in the γ’-phase behind the leading partials. This view is also a basic element of later publications where the segregation of atoms to planar faults is considered [17,51-54]. Recent work on the formation of planar faults in Co- and Co-

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Ni based superalloys with γ/γ’-microstructures [55-57] have provided new interesting results on the physical nature of planar faults in the ordered L12 phase. Planar faults in Co-base SX have lower energies than in Ni-base SX [e.g. 56]. In the present work we investigate a well-characterized Ni-base SX [14,58,59] which was tensile creep deformed in a precise [001] orientation [14, 22]. We use diffraction

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contrast TEM analysis to identify γ-channel dislocation reactions which precede γ’phase cutting. In addition to conventional g⋅b analysis (g – reciprocal lattice vector of reflecting planes, b – Burgers vector of dislocation under observation) we employ large-angle convergent beam electron diffraction (LACBED) for the determination of

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complete Burgers vectors [57,60]. We apply stereo scanning TEM (STEM) to consider the spatial orientation of linear and planar faults [61-63]. We extent a

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previous discrete dislocation model (DDM) [23,25] to rationalize dislocation structures which were observed using TEM and to explain dislocation reactions which explain the nucleation of <112> ribbons.

Material and Methods Material, creep testing and microstructure: The Ni-base SX ERBO1/C (CMSX-4 type) is investigated in the present work [58]. The chemical composition and the partitioning of elements between prior dendritic and interdendritic regions and between the γ- and the γ’-phase have been established [58,59]. In the as heat-treated 4

ACCEPTED MANUSCRIPT condition, the volume fractions of the γ’- and γ-phase are close to 75% and 25% respectively. The scatter in γ-channel widths and γ’-particle sizes has been documented [14]. In the present work we investigate a material state which was crept at 750°C and 800 MPa using precisely [001] oriented miniature tensile creep specimens [14,22]. [001] tensile creep experiments were interrupted after 0.4 and 1%

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strain, where the dislocation density in the horizontal γ-channels is significantly higher than in the vertical channels [14]. In the present work {111} TEM foils were taken from the creep specimens. The 1% specimen was investigated in a Jeol JEM-2100F. Most details describing our TEM method have been published elsewhere

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[14,57,58,60-63]. After 1% strain the dislocation density in the γ-channels was so high that it was difficult to distinguish individual dislocation processes. Therefore a second creep test was interrupted after 0.4%. The corresponding specimen was

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investigated in a double-aberration-corrected FEI Titan Themis³ 300 operating at an acceleration voltage of 300 kV. Annular dark field scanning TEM (ADF-STEM) was performed on low indexed Kikuchi bands of type {111} in order to obtain diffraction contrast information from linear and planar defects [57,64]. LACBED was then applied in the nano probe mode to characterize selected planar defects.

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TEM tilt experiments after 1% creep deformation: A series of sixteen tilt experiments was performed. In each tilt position a two-beam diffraction contrast g (reciprocal lattice vector) was established using a small positive deviation parameter w given by the extinction distance ξg and the magnitude of the deviation vector s [65-67]. The

(1).

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w = ξg s

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deviation parameter w is obtained as

The deviation vector s can be determined using s=

x λ ⋅ R d2

(2),

where x is the distance of the outward shift of the Kikuchi lines, R is the measured distance between the spots created by the transmitted and diffracted beams, λ is the wavelength of the electrons and d is the lattice spacing of the reflecting planes. Table 1 compiles all g-vectors and the experimental deviation parameters w which were used to analyze the 1% creep specimen. 5

ACCEPTED MANUSCRIPT Figure 1 presents four TEM micrographs from a characteristic region in 1% creep specimen. The directions of the g-vectors (black arrows) and the orientations of the TEM-foils are indicated. The area of interest contains a horizontal channel h1, Figure 1a. There are five vertical channels, v1 to v5 in Figure 1b. In Figures 1a to d fourteen individual dislocations (1 – 14) are highlighted by small arrows. Five planar faults are

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highlighted with f1, f2, f3, f4 and f5 in Figures 1c and d. Six specific locations are numbered from 1 to 6 for later reference. In case of the 1% specimen, Burgers vectors of dislocations are determined based on effective visibility/invisibility criteria [65-67]. Contrasts of partial dislocations with Burgers vectors bP, which limit stacking

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faults, are of special interest [68].

LACBED analysis after 0.4% creep deformation: In order to directly determine the

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absolute sign and the magnitude of Burgers vectors of partial dislocations associated with planar faults, we apply the LACBED method [60,69,70]. A convergent electron beam with a large convergence angle is focused to the eucentric height. This results in overlapping diffraction discs in the back focal plane. The TEM foil is then lowered from the eucentric position, which creates a regular pattern of diffracted beams in the first image plane. Using a small selected area aperture allows to either select the

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direct beam or the diffracted beam. One can thus observe individual diffraction discs in the final diffraction pattern. As has been described in the literature [60,69,70], these LACBED discs not only show patterns of Bragg lines, but also contain the shadow image of the illuminated sample area. As first demonstrated by Cherns and

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Preston [71] the Burgers vectors of dislocations can be unambiguously determined by evaluating the contrast of the Bragg lines which intersect the traces of dislocation

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lines with line direction u. The presence of a dislocation introduces distinct maxima and minima in the Bragg lines. Using the Cherns and Preston rule [69-71] the Burgers vector of a dislocation can be deduced from the number of maxima. Stereo TEM: Stereo TEM allows to appreciate the true three dimensional nature of

dislocation and planar fault configurations. In the present work we apply the stereo TEM procedure described by Agudo et al. [61], where the two micrographs of a stereo pair are combined into one anaglyph. When these anaglyphs are viewed with colored glasses (left eye: red, right eye: cyan), one obtains a realistic 3D impression of the microstructure. For the present work four stereo images were made, which

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ACCEPTED MANUSCRIPT were taken at constant gs (marked with an “A” in Table 1) in angular distances between 10 and 16°. 2D Discrete Dislocation Dynamics Modelling (2D DDD): In the present work we use a

2D DDD model, which considers the interaction between two 60° dislocations of types 1/2[01 1] and 1/2[10 1] in the γ/γ’-microstructure. Figures 2a to c illustrate three

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modelling stages which show a central horizontal γ-channel (view direction: [1 10] ). Figure 2a represents an initial state, where five 1/2[01 1] loops have deposited 60° dislocation segments at the upper and lower γ/γ’-interfaces. These segments are

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projected end-on. We fix the outer four black segments and allow the six blue segments to move along the interfaces by combined climb and glide processes. The loading direction σa and the directions of Peach-Koehler glide and climb forces Fg

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and Fc are indicated. The Peach-Koehler force F acting on a dislocation with a Burgers vector b and a line direction e is obtained as [72]

F = b ⋅ ( σa + σm + σd ) ×e

(3).

It is governed by the overall stress state σ which results from the superposition of

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three stress terms, the applied stress σa, the misfit stress σm and a stress term which reflects the presence of other dislocations σd [e.g. 23]. The Peach-Koehler force F from Equation 3 is readily decomposed into its glide and climb components [72] (4),

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Fg = (F ⋅ eg ) eg and Fc = (F ⋅ ec ) ec

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where eg and ec represent unit vectors in the climb and glide directions. In Figure 2b two interface dislocations move by a combined movement of glide and climb along the γ/γ’-interface. The glide and climb velocities vg and vc are linear combinations of the Peach-Koehler forces and of the kinetic constants Cg and Cc: v g = C g Fg and v c = Cc Fc

(5).

The two kinetic constants Cg and Cc are calculated applying a thermodynamic extremum principle [73, 74], which maximizes the rate of elastic strain energy dissipation as the mixed dislocation segments move. They are obtained as

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ACCEPTED MANUSCRIPT Cg =

2 Ag Asg

Ag + Asg + ( Ag − Asg ) cos 2 β

and Cc =

2 Ac Asc

Ac + Asc + ( Ac − Asc ) cos 2β

(6).

Ag and Ac in Equation 6 represent the kinetic parameters which describe the glide and climb kinetics of a pure edge dislocation segment, while Asg and Asc are the corresponding parameters for the motion of screw segments. β is the angle between

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b and the dislocation line direction e. In our calculation we assume that glide is 10

times faster than climb (Ag = 10 Ac). The climb parameter Ac is related to a representative diffusion coefficient and the temperature as [72] D0 exp[-Q/(kT)] Ω

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Ac (T) =

b2 kT

(7).

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In Equation 7, D0 is the pre-exponential factor of the diffusion coefficient. The exponential term describes the temperature dependence of the diffusion coefficient (Q – activation energy, k and T have their usual meanings). Ω is the atomic volume and b is the magnitude of the Burgers vector [75]. For the present calculation we use diffusion data for Ni as reported in [75]. We moreover assume that the motion of screw dislocations in glide direction is 1.5 times faster than that of edge dislocations

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(Asg = 1.5·Ag) and that their motion in climb direction is two times faster (Asc = 2·Ac). The input parameters which were used in the present work are listed in Table 2. In

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0 0 0  σa =  0 0 0  0 0 S  

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the present work we express the applied stress S by

Two explicit misfit stress states are considered, σ mH the misfit stress in horizontal channels and σ mP , the misfit stress in γ’-particles:

σ mH

0  0 0   −M 0  M / 10     =  0 −M 0  σ mP =  0 M / 10 0   0  0 0 M / 10  0 M / 10   

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(9).

ACCEPTED MANUSCRIPT Here M is the magnitude of the misfit stress [16]. The parameters which are required for the micromechanical calculations are listed in Table 3. The present model also addresses cutting processes, where planar faults are created. The fault energies considered in the present work are listed in Table 3. These energies represent

Microstructural Results

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force/length-values and can be directly interpreted as Peach-Koehler forces.

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We first consider the TEM results which were obtained for the 1% creep specimen. To the right of asterisk 1 in Figure 1a, a planar fault appears to have just nucleated

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by a dislocation entering the γ’ particle from a vertical channel. Asterisk 2 in Figure 1a

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highlights a planar fault which lies on the 1 1 1 plane. Where this inclined defect ends, it appears to be connected to γ-channel dislocations. The asterisks 3 and 4 in Figure 1b show that the planar faults in the upper and lower γ’-particles are effectively invisible. At the location of asterisk 5 in Figure 1c, several planar faults above each

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other make a precise defect analysis difficult. In Figure 1d, asterisk 6 marks a channel, where γ/γ’-interfaces are edge-on. In the TEM micrographs of Figure 1, fourteen dislocations are highlighted (1 to 14). In Figure 1a, a horizontal channel segment is highlighted as h1. Five vertical channel segments (v1 to v5) are marked in

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Figure 1b. Five planar faults are highlighted in Figures 1c and 1d, referred to as f1 to f5. Figure 3 shows a 3D anaglyph of the region shown in the TEM micrographs of Figure 1. The anaglyph was constructed using the two stereo micrographs taken at

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(

g3A and g4A (both: 202 , see Table 1). Using colored glasses one can appreciate the

spatial arrangement of the elements of the microstructure [61]. In the anaglyph of Figure 3 the planar faults f1, f2, f3 and f4 which are clearly visible in Figure 1a, c and d are out of contrast. But, in combination with the results shown in Figures 1a, c and d, Figure 3 allows to conclude that they are parallel to the (111) plane. In contrast, the inclined faults like the one marked with *2 in Figure 1a, are clearly visible in the anaglyph of Figure 3. The anaglyph also reveals that in channel v2 (marked in Figure 3b), the two dislocations 11 and 12 glide in parallel (111) planes at different heights, dislocation 9

ACCEPTED MANUSCRIPT 11 being closer to the bottom surface of the TEM foil. Dislocations 7, 8 and 9 have line directions which are close to [1 1 0] (classical deposition scenario). In contrast, dislocation segments 2, 3 and 4 (as defined in Figure 1a) appear to have moved away from their original deposition direction. This represents direct experimental evidence for the contribution of climb processes to the evolution of dislocation

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substructures in γ/γ’-interfaces. The anaglyph shown in Figure 3 also provides important information on dislocation segments 13 and 14. First, the curvature of the two dislocations allows to conclude that they were frozen in during a downward motion. Second, they both glide in the

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same (111) plane at the same height of the TEM foil. Third, dislocation 14 spreads continuously from the left γ’-particle through the intermediate γ-channel into the right

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γ’-particle. In contrast, dislocation segment 13 is confined to the γ-channel. The positions where its segments are deposited in the γ/γ’-interfaces can be clearly recognized.

The results of the contrast analysis performed in the present work are presented in Tables 4a and b. For reasons of space we list the results for dislocations 1 to 7 from

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Figure 1 in Table 4a. Table 4b shows the results for dislocations 8 to 14. All other columns report the visibilities (full visibility: +, invisibility: -, residual contrast: res). In addition to the visibility results, we list the scalar products between operating g and b which best match the experimental data. Dislocations d1 to d7 listed in Table 4a have

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bs of ± 1/2[01 1]. Dislocations 8 and 9 from Table 4b have the same b. Dislocations

11 and 12 from the same microstructural location, have bs of ± 1/2[10 1] .

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We refer to dislocations with bs of ± 1/2[01 1] and ± 1/2[10 1] as dislocation families 1 and 2. Figures 1 and 3 provide direct evidence that the dislocation segments 7, 8 and 9 (family 1) and the dislocation segments 11 and 12 (family 2) were deposited at the γ/γ’-interfaces by glide on the same set of (111) planes. Consequently, glide on the common set of (111) planes, involving dislocations from both families, would leave behind long segments parallel to  110  in the horizontal channel h1. Dislocations 13 and 14 from Table 4b, with bs of 1/6[1 1 2] and 1/3[1 1 2] most probably have formed by reactions between dislocations from dislocation families 1 and 2. We take a closer look at dislocations 13 and 14 which glide in the same plane, Figure 3. 10

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We show the two dislocations 13 and 14 in Figure 4 at a higher magnification. The Burgers vector of dislocation 14 has been determined as 1/3[1 1 2] . This dislocation bridges the γ-channel between two γ’-particles. In the γ’-particles it limits planar faults. The planar fault in the right γ’-particle has been highlighted as f1 in Figure 1c. The γ-

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channel dislocation 13 has a Burgers vector of 1/6[1 1 2] and follows d14. In Figures 1c and d it appears as if d14 and d13 represent leading and trailing dislocation segments. However, while the leading dislocation d14 continues from the γ’-particle to the left through the vertical γ-channel v4 (see Figure 1b) into the γ’-particle to the

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right, the trailing dislocation d13 is fully confined to the γ-channel.

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The TEM image in Figure 4 taken under a g of 1 1 1 suggests, that in the γ-channel, dislocation d14 splits up. It shows the vertical γ -channel v4 at a higher magnification with a focus on the dislocations d13 and d14. Figure 4a identifies the planar fault f1 and also highlights the dislocation segment d13 with its b of 1/6[1 1 2] . The black dashed lines in Figure 4b highlight the position of the γ-channel. The forefront of dislocation 14 (Figure 1d) is shown as white dashed line in the two γ’-particles.

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Where it transects the γ-channel, a dark half-moon shaped contrast suggests the presence of a stacking fault, as indicated by two dashed line segments. The two γchannel dislocations with bs which add up to the b of d14 travel in close distance. The stacking fault which gives rise to the half-moon contrast can be created by either

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a leading 1/6[1 2 1] or a leading 1/6[2 11] dislocation segment. Further work is required to identify which of the two configurations describes the correct splitting mode of the

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overall 1/3[1 1 2] dislocation. It was not possible to determine the fault vector of the planar fault f5 highlighted in Figure 1d, because it is too small. However, there are two invisibilities for the planar faults f1 to f4 as documented in Figure 1b and Figure 3, suggesting that their fault vectors are of type 1/3(111) . The LACBED analysis shown in Figure 5 was performed for the specimen after a creep deformation of 0.4%. Figure 5a shows a STEM ADF overview image which

(

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was taken at a g of 11 1 . The projected [001] arrow marked with a σ indicates the loading direction during creep. The region limited by the white dashed rectangle in Figure 5a is shown at a higher magnification in Figures 5b and c. Figure 5b 11

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(

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represents a weak beam dark field image taken for a g-3g condition 202 . Figure 5c was taken at a g-vector of

(220) ,

where the planar faults and the channel

dislocations are effectively invisible. Figures 5d and e show the region of interest (black dotted rectangular region in Figure 5a) at a higher magnification. The STEM image in Figure 5d clearly shows that the planar fault in the γ’ phase extends into the

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γ-channel. The complementary conventional weak beam dark field image, Figure 5e, shows that the γ channel stacking fault is associated with 3 dislocations, two leading dislocations (1 and 2) and one trailing dislocation (3). In the lower part of the image, channel dislocations 1 and 2 approach each other to form a narrow band inside the γ’

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precipitate. The schematic illustration in Figure 5f summarizes these observations and introduces a color coding for the 3 dislocations, red and green for the two leading

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dislocations and blue for the trailing dislocation. This color coding is used in Figures 5g and h, which show the results of the LACBED analysis. Figures 5g and h show characteristic LACBED images, which combine microstructural results (shadow images) with diffraction information. Figure 5g addresses the two dislocations inside the γ’ precipitate which appear in weak contrast and are highlighted by dotted red and dotted green lines. LACBED provides evidence for the lattice disturbances, resulting

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from the strain fields of dislocations which cause characteristic disruptions. These

(

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characteristic splittings can be recognized in Figure 5g for the solid blue 044 ,

(

(

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orange 224 and violet 426 Bragg lines. Following the Bragg lines from the left to

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the right and approaching the intersection with the dislocation lines, one finds intensity minima and maxima. When crossing the dislocations (marked by red and

(

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green dotted lines), the blue Bragg line 044 splits into four subsidiary intensity maxima (n=+4), the orange Bragg line into four (n=-4) and the violet Bragg line into six (n=-6). Following Cherns and Preston [71] we take the direction u of the dislocation lines into account, consider the sign of the excitation error s and consider the sense of the Bragg line splitting (right or left). We then calculate the three scalar

(

)

products g ⋅b = n : 044 ⋅b = +4 ,

(224)⋅b = −4 , (426 )⋅b = −6

and thus obtain the

Burgers vector as + 1/3[1 1 2] for the pair of dislocations inside the γ’ precipitate. The three gs are linearly dependent which can be seen from the fact that they are all contained in the (111) glide plane. In fact, evaluation of either two splittings are sufficient to determine the complete Burgers vector including its sign and magnitude. The third splitting then confirms the consistency of the LACBED analysis. The 12

ACCEPTED MANUSCRIPT resulting b of + 1/3[1 1 2] is in agreement with the invisibility of the dislocation pair in

(

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the CTEM image of Figure 5c, taken for g= 2 20 . Figures 5h and i provide information on the blue γ-channel dislocation. The splittings are apparent when following the solid red and turquoise lines. Performing the same type of analysis we find that the red Bragg line

(224)

splits into two intensity maxima (n=-2),

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(224)⋅b = −2 and the turquoise Bragg line (242) splits into one intensity maxima (n=+1), ( 242) ⋅ b = +1. Since b must be contained in the (111) glide plane, the LACBED analysis uniquely identifies the Burgers vector as b = 1/6[1 1 2]. This is

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consistent with the invisibility of the dislocation contrast due to g ⋅b = 0 in the CTEM

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image taken at g = 2 20 in Figure 5c. On the left side of the γ-channel, the three

(

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dislocations with a total b of 1/2[1 1 2] come very close to each other. Indeed the

)

turquoise Bragg line in Figure 5i 242 splits into three subsidiary intensity maxima (n=+3), as expected for a b of 1/2[1 1 2] .

Figure 5f summarizes the results of our weak beam / LACBED analysis. The two

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leading dislocations form a pair in the γ’-precipitate with a total b of + 1/3[1 1 2] . Our analysis suggests a small separation of the two dislocations. However, a dedicated high resolution analysis in edge-on geometry is required to determine the nature of the splitting. In the γ-channel the two dislocations show a larger separation as clearly

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revealed by the weak beam dark field image in Figure 5e. However, our analysis does not allow to precisely determine the individual bs. The trailing partial (blue)

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complementing the SF ribbon possess a Burgers vector of 1/6[1 1 2] and completely remains in the γ-channel. Based on our experimental observations and on energetic considerations, the following cutting scenario is proposed: (1) Two channel dislocations approach the γ’-precipitate on the (111) plane ( 1/2[0 1 1] and 1/2[1 01] ). (2) When the leading γ-channel dislocation 1/2[0 1 1] enters the γ’-precipitate, an APB is created. (3) The trailing channel dislocation splits into two Shockley partials creating an ordinary SF in the γ –channel. (4) The first Shockley partial dislocation enters the γ’-precipitate and converts the APB into a lower energy SISF. In this scenario, a leading 1/2[0 1 1] and a trailing 1/6[2 11] partial dislocation move separately in the γchannel. They combine and shear the γ’-phase as a + 1/3[1 1 2] dislocation. In the γ13

ACCEPTED MANUSCRIPT channel the two dislocations produce a SF, which is removed by a dislocation with a b of 1/6[1 1 2] , which remains in the γ-channel at the γ/γ’-interface.

Modelling Results

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Figure 6 shows Peach-Koehler force maps which act on a probing dislocation as explained in [25]. In image center there is an interface dislocation of type 1/2[01 1] . Peach Koehler forces act on a probing dislocation of type 1/2[10 1] . Directions of

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positive glide and climb forces are indicated by arrows. In the left column of Figure 6 the external and misfit stresses are considered. The right column of Figure 6 shows results which were obtained when dislocation stress fields from other γ-channel

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dislocations are superimposed, considering the γ-channel dislocation structure presented in Figure 2c. Figures 6e and f integrate the results obtained for glide and climb forces. They show blue (glide) and red (climb) contours, which indicate locations where Peach-Koehler forces are zero. Regions marked with “+” and “-” signs indicate the directions into which the approaching 1/2[10 1] dislocation is driven

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with respect to the directions defined by the reference arrows. The presence of other channel dislocations has only little effect on the Peach-Koehler glide forces (blue contours hardly change). In contrast, it has a significant influence on climb. It is interesting to consider the blue segment which connects the points “1” and “2” in

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Figures 6e and f. When the probing dislocation 1/2[10 1] reaches this segment, it gradually moves by coupled slow climb and fast glide steps until it reaches point 1. In the context of the present work it is important to highlight that point 1 is located on the

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(111) plane which is also the glide plane of the interface dislocation 1/2[01 1] . The two 60° γ-channel dislocations can not only reach a common (111) glide plane but also come as close to each other as 5 nm. We now show how the lower red dislocation segment from Figure 2c approaches a central interface dislocation, Figure 7. The 1/2[10 1] dislocation is driven by PeachKoehler glide and climb forces which are governed by the overall local stress state in field A of Figure 2c. At point 1 in Figure 7, the dislocation approaches the 1/2[01 1] interface dislocation driven by a high glide force. Climb forces push the dislocation towards the common (111) glide plane. At point 2 (Fg=0), the path of the dislocation 14

ACCEPTED MANUSCRIPT crosses the trace of the (111) plane. It then moves towards point 3 by a combined climb and glide motion. On its way from point 2 to 3, the approaching dislocation circumvents the interface dislocation and reaches the γ/γ’ interface at point 3. Here the acting glide force is not high enough to push it into the γ’ phase, where it would have to overcome the counteracting APB force. Therefore it moves along the γ/γ’

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interface towards point 4. Figure 8a describes the evolution of two distances, as the 60° dislocation 1/2[10 1] travels from points 1 to 2 in Figure 7. The red curve represents the distance between the moving dislocation and the common (111) glide plane. The blue curve shows the

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distance between the moving dislocation and the 1/2[01 1] interface dislocation measured along the glide direction in the glide plane. The dashed vertical line in

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Figure 8a indicates the time where the approaching dislocation reaches point 2 in Figure 7. Note that there is a short time interval where both dislocations are located close to each other (5 nm distance) on the common (111) glide plane. The black curve in Figure 8b shows, how Peach-Koehler glide forces change when we displace the dislocation 1/2[10 1] along the common (111) glide plane, where point 2 is a

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stable position. We now consider the possibility that at point 2, the approaching dislocation splits into two stacking fault coupled Shockley partials. The bs of the leading and trailing partials are 1/6[2 1 1] and 1/6[11 2 ], respectively. Figure 8b shows that after splitting, the two partials also reach equilibrium positions close to the γ/γ’interface (leading partial: blue curve, trailing partial: red curve). It is important to point

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out that when the approaching 1/2[10 1] dislocation splits up into two Shockley

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partials, the elastic strain energy decreases by about 25%, therefore this process can occur spontaneously.

Results of detailed calculations (not presented in this study) show that similar stable decompositions into Shockley partials can occur on (111) planes adjacent to the (111) plane of the interface dislocations 1/2[01 1] . The calculations also show, that splitting on more distant (111) planes is unlikely, because the resulting configurations are not stable and no stable stacking faults can be formed. In Figure 9 the two 60° dislocations required for t he nucleation of a planar fault are referred

to

as

1

(interface

dislocation, 1/2[01 1] ) 15

and

2

(approaching

ACCEPTED MANUSCRIPT dislocation, 1/2[10 1] ). In Figure 9a the approaching dislocation 2 has reached position 2 in Figure 7 after 103.5 s. We then reset the time to 0. The approaching dislocation 2 splits up into two partials, the leading partial 2a ( 1/6[2 1 1] ) and the trailing partial 2b ( 1/6[11 2 ]), Figure 9b. After 1.1 s the interface dislocation 1 has been pushed into the γ’-particle, where it cannot penetrate very far, due to the counteracting APB force,

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Figure 9c. In Figure 9d, after 1.545 s, the leading partial 2a has reached the γ/γ’interface. It only takes as little as 30 ms, before it enters the γ’-phase, where it immediately joins dislocation 1 and forms a 1/3[11 2 ] superpartial, Figure 9e. The 1-2a configuration represents a leading superpartial. We keep in mind that our model is

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based on linear elasticity and can therefore not account for processes which are governed by the atomistic nature of the dislocation core. Dislocations 1 and 2a do not

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merge but remain APB coupled in a very small distance of 0.46 nm. Once the SISF has formed, the leading superpartial 1-2a can move into the γ’-phase and the partial dislocation 2b has reached the γ/γ’-interface. The configuration after 2.1s is shown in Figure 9f. At this stage, the Peach-Koehler glide force acting on the Shockley partial 2b is not yet high enough to push it into the γ’-phase.

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In order to be able to appreciate the sequence of events compiled in Figure 9, it is important to document (i) how the dislocations involved in the process change their positions with time and (ii) how the corresponding Peach-Koehler glide forces evolve, Figure 10. In Figure 10a, a black horizontal line at ordinate 0 marks the position of

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the γ/γ’-interface. The positions of dislocations 1, 2a and 2b are shown as blue, yellow and green curves. At time 0, dislocation 1 sits at the interface while the two

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Shockley partials occupy their equilibrium positions in the γ-channel. Dislocation 1 enters the γ’-particle as dislocations 2a and 2b approach. When dislocation 2a has reached the interface (first arrow a), it quickly joins dislocation 1. As dislocations 1 and 2a jointly shear the γ’-phase, dislocation 2b approaches the γ/γ’-interface, which it reaches after the time indicated by the arrow b. Figure 10b shows how the PeachKoehler glide forces, which act on dislocations 1, 2a and 2b, evolve with time. The dashed horizontal line indicates the force which is required to create an APB. Figure 10b shows that this high counteracting APB Peach-Koehler force can be overcome, when interface dislocation 1 enters the particle. Once in the particle, dislocation 1 only feels a small forward Peach-Koehler glide force, which results from 16

ACCEPTED MANUSCRIPT the superposition of all stress components, including the counteracting APB force. Up to 1.545s, all three dislocations see small positive forward glide forces. As soon as dislocation 2a reaches the interface, the strong tendency to eliminate the APB pulls into the γ’-phase towards dislocation 1. This is reflected in a strong sudden increase of Peach-Koehler glide forces acting on dislocations 1 and 2a. This also strongly

superpartial moves on.

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Discussion

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increases the driving force on dislocation 1, which moderately decreases as the 1-2a

Link between γ -channel dislocation activity and γ’-phase cutting: Sass and Feller-

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Kniepmeier [41] observed that the reaction between a 60° and a screw dislocation can nucleate a planar fault. They concluded that it is unlikely that two 60° dislocations with different Burgers vectors can do the same. The results of the present work show that this is possible, Figure 5. Our current results are not in line with the Kear scenario [30], where it is assumed that three fault coupled γ-matrix partial

dislocations approach the γ/γ’-interface. This is not in line with what is actually

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observed. During the early stages of creep, leading 1/2<110> segments move on {111} planes into narrow γ-channels and deposit 60° dislocation segments in t he γ/γ’interfaces [23]. The results obtained in the present work clearly prove the presence of such 60° interface dislocations, Figure 1. The scen ario proposed in the present work

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also differs from the Decamps scenario [39]. Our results are in line with their findings that 1/2<110> dislocations govern γ-channel plasticity and that early in the process, a

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1/2<110> dislocation cuts into the γ’-phase. However, our results suggest that no extended APB ribbons can form, because the Peach-Koehler force, which is high enough to push the dislocation into the γ’-phase, is not sufficient to let the APB grow. Our model suggests, that the formation of the leading 1/3[11 2 ] dislocation involves the reaction between a dislocation 1 (Burgers vector: 1/2[01 1] ) and a partial dislocation 2a (Burgers vector: 1/6[2 1 1] ), which results from a splitting of the second

[

]

[

]

1/2 10 1 channel dislocation. The observation of a trailing 1/6 1 1 2 dislocation which

is left behind at the γ/γ’-interface supports this scenario, Figures 4 and 5. Rae and Zhang [47] pointed out, that it is not clear whether for dislocations to combine efficiently they need to be on the same slip plane, as was assumed by Ma et al. [77]. 17

ACCEPTED MANUSCRIPT The results of our TEM investigations and our DDD model help in this respect. They show that one dislocation with a b of 1/2[10 1] can approach another 1/2[01 1] dislocation and reach its (111) glide plane by a combined motion of glide and climb, Figures 7 and 8. Opening and closing of the window of the Rae-window [8, 14]): Rae et al. [44,46,47]

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suggested that there is a narrow window of opportunity for the onset of intense primary creep. This requires the presence of a sufficient number of 1/2<110> dislocations with suitable Burgers vectors for the nucleation of <112>{111} ribbons [44,46,47] to occur. However, there should not be too many dislocations in the γ-

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channels, for dislocation propagation to be impaired [44,46,47]. As the dislocation density in the γ-channels increases, the Rae-window eventually closes. The

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nucleation of a full <112>{111} dislocation ribbon starts with the nucleation of a 1/3<112> dislocation. One can interpret the segments which connect the points 1 and 2 in Figures 6e and f as attractors from where the nucleation reaction shown in Figure 9 starts. A sufficient number of 60° interfa ce dislocations are needed to provide a sufficient number of such attractors. As the density of interface dislocations increases in the early stages of primary creep [14], the number of these attractors

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increases. This elementary process accounts for the opening of the Rae-window. The results of our DDD simulation suggest that as the number of interface dislocations increases further, the extension of attractor segments shortens (Figures 6e and f), resulting in a lower probability for the nucleation sequence shown in Figure 9 to

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commence. When the tension and compression hydrostatic stress fields of the interface dislocations are taken into account, they superimpose with similar

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components due to γ/γ’-misfit. The superposition reduces the climb force acting on the approaching 1/2[10 1] dislocation which further results in the shift of the zero climb force contour towards the point 1 and thus in the shortening of the attractor segment. An increasing dislocation density in the γ-channels eventually leads to the closing of the Rae-window. Reordering in the non-compact 1/3<112> dislocation core: Kovarik et al. [48] have

clearly shown that the creation of a SISF by glide of a 1/3<112> super dislocation can be associated with diffusion mediated re-ordering. They showed that the leading 1/3<112> segment which limits the SISF in the γ’-phase is formed as the result of coordinated movement of three Shockley partials on two adjacent {111} planes. Our 18

ACCEPTED MANUSCRIPT diffraction contrast TEM results do not allow to resolve these fine details and this mechanism is not addressed in our model. It would modify the values of the kinetic constants Cg and Cc for the dislocation motion in the γ’-phase. On nucleation, Co-based SX and grooves in γ/γ’-interfaces: Full <112> ribbons must

nucleate [44]. But one must keep in mind that this nucleation process results from a

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chain of individual nucleation events, starting with the nucleation of a leading 1/3<112> dislocation as described in Figure 9. In the case of Co-based single crystal superalloys with γ/γ’-microstructures, much lower fault energies were reported [5557,78,79]. Therefore it is not surprising that other planar fault structures were

appearance of extended APBs in Co-based SX.

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observed [65,57,79,]. Especially, a much lower APB energy rationalizes the

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On the difficulty to observe the nucleation of planar faults: The presence of extended

planar faults in the γ’-phase of Ni-base SX was observed in the present work and has been frequently reported in the literature. In contrast, nucleation events have only been rarely reported, like for example by Sass and Feller-Kniepmeier [41]. In the present work, a nucleation event has been observed in Figure 1a close to asterisk 1, and another nucleation event was analyzed in Figure 5. In our model, the nucleation

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event starts when the configuration of Figure 9a is reached. From there on it only takes 2s to establish the configuration in Figure 9f, where an extended SISF has formed in the γ’-phase. Many such events must occur to rationalize low temperature

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and high stress creep [14]. However, the fault ribbons move on quickly and therefore

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nucleation events are difficult to observe.

Summary and Conclusions

The present work addresses the link between γ-channel dislocation plasticity and γ’phase cutting under conditions of low temperature (T=750°C) and high stress ( σ=800 MPa) creep of single crystal Ni-base superalloys. From the results obtained in the present work the following conclusions can be drawn: (1) Using diffraction contrast TEM, two families of 60° γ-channel dislocations with Burgers vectors of 1/2[01 1] and 1/2[10 1] were identified. These two dislocations are required to form the first part of a <112>{111} dislocation ribbon. 19

ACCEPTED MANUSCRIPT (2) A SISF coupled pair of a leading 1/3[11 2 ] superpartial in the γ’-phase and a trailing

[ ]

1/6 11 2 dislocation at the γ/γ’-interface were identified (diffraction contrast TEM and

LACBED). (3) It was observed that the 1/ 3[1 12] superpartial was continuous across a γ-

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channel between two γ’-particles. In the γ-channel this dislocation dissociates into two dislocations. Even though the individual bs could not be determined, the two dislocations are likely an octahedral 1/2[0 1 1] dislocation and a 1/6[2 11] Shockley partial dislocation. While the octahedral dislocation can glide through the γ-phase without changing its stacking sequence, the Shockley partial is SF-coupled to another

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trailing 1/6[1 1 2] Shockley partial dislocation. High-resolution TEM analyses of the

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dislocation core structures are necessary to identify the exact physical nature of the fault coupled dislocations in the γ-channel.

(4) In the present work we use a 2D discrete dislocation dynamics model, which builds up on our TEM observations and helps to bridge the gap between the understanding of γ-channel dislocation plasticity and SISF-controlled γ’-phase cutting. It is based on realistic assumptions concerning the overall stress state and the role of

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planar fault energies. Based on the model results we propose a new scenario. The model shows, that one arriving 1/2<110> 60° channel dislocation can approach another 1/2<110> 60° interface dislocation by a com bined motion of glide and climb.

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It can come as close as 5 nm on a common (111) plane. It can then spontaneously dissociate into two Shockley partials. This represents the critical nucleation event of

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the first part of a <112> - ribbon. (5) The new mechanism proposed in the present work provides insight into why the Rae-window [14] opens and closes in the early stages of low temperature and high stress primary creep. The window opens because as the dislocation density moderately increases, attractors are created where two 60° dislocations can approach each other by glide and climb and reach a common glide plane. The window closes because a strong increase of the dislocation density reduces the size of attractors and impedes dislocation motion.

Acknowledgement 20

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References

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XW, AK, CS and GE acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) through project A2 of SFB/TR 103. YME and ES acknowledge funding by DFG through project A7 of SFB/TR 103. AK acknowledges funding by the Zentrum für Grenzflächendominierte Hochleistungswerkstoffe (ZGH) of the Ruhr-Universität Bochum. XW acknowledges funding through the IMPRS SurMat at MPIE Düsseldorf. AD acknowledges funding by the CSF through project 14-22834 S. The authors acknowledge help and fruitful discussions from/with M.Sc. Malte Lenz and Dr.-Ing. Julian Müller from IMN FAU.

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ACCEPTED MANUSCRIPT base superalloys, Acta Mater. 113 (2016) 335-349. [58] A.B. Parsa, P. Wollgramm, H. Buck, C. Somsen, A. Kostka, I. Povstugar, P. Choi, D. Raabe, A. Dlouhy, J. Müller, E. Spiecker, K. Demtröder, J. Schreuer, K. Neuking, G. Eggeler, Advanced Scale Bridging Microstructure Analysis of Single Crystal Ni-Base Superalloys, Adv. Eng. Mater.17 (2015) 216-230.

RI PT

[59] V. Yardley, I. Povstugar, P.P. Choi, D. Raabe, A.B. Parsa, A. Kostka, C. Somsen, A. Dlouhy, K. Neuking, E. George, G. Eggeler, On thermodynamic equilibria and small and large scale chemical and microstructural heterogeneities in single crystal Ni-base superalloys, Adv. Eng. Mater. 18 (2016) 1556-1567.

SC

[60] J. Müller, G. Eggeler, E. Spiecker, On the identification of superdislocations in the γ’-phase of single-crystal Ni-base superalloys – An application of the LACBED method to complex microstructures, Acta Mater. 87 (2015) 34-44.

M AN U

[61] L. Jácome Agudo, G. Eggeler, A. Dlouhy, Advanced scanning transmission stereo electron microscopy of structural and functional engineering materials, Ultramicroscopy, 122 (2012) 48–59. [62] A.B. Parsa, P. Wollgramm, H. Buck, A. Kostka, C. Somsen, A. Dlouhy, G. Eggeler, Ledges and grooves at γ/ γ’ interfaces of single crystal superalloys, Acta Mater. 90 (2015) 105-117. [63] A. Dlouhy, G. Eggeler, Superdislocation line directions in γ’-particles after shear creep deformation of superalloy single crystals, Pract. Metallogr. 33 (1996) 629-642.

TE D

[64] P.J. Philips, M.C. Brandes, M.J. Mills, M.De Graf, Diffraction contrast STEM of dislocations: imaging and simulations, Ultramicroscopy 11(2011) 1483-1487. [65] P. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.H. Whelan, Electron microscopy of thin crystals, Butterworths, Washington, 1965.

EP

[66] J.W. Edington, Practical electron microscopy in materials science, Van Norstrand Reinhold Company, New York, 1976.

AC C

[67] D.B. Wiliams, C.B. Carter, Transmission Electron Microscopy – II Diffraction, Plenum Press, New York, 1996. [68] J.M. Silcock, W.J. Tunstall, Partial dislocations associated with NbC precipitation in austenitic steels, Philos. Mag. 10 (1964) 361-389. [69] J.P. Morniroli, Large angle convergent beam electron diffraction, Institute of Physics Publishing, Paris, 2004. [70] J.P. Morniroli, Large-angle convergent-beam electron diffraction applications to crystal defects, CRC Press, Boca Raton, Florida, 2004. [71] D. Cherns, R. Preston, Convergent beam diffraction studies of interfaces, defects and multilayers, J. Electron Microsc. Techn. 13 (1989) 111-122.

25

ACCEPTED MANUSCRIPT [72] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., Krieger Publishing Company, Malabar, Florida, 1992. [73] F.D. Fischer, J. Svoboda, H. Petryk, Thermodynamic extremal principles for irreversible processes in materials science, Acta Mater. 67 (2014) 1-20.

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[74] T. Zalezak, J. Svoboda, A. Dlouhy, High temperature dislocation processes in precipitation hardened crystals investigated by a 3D discrete dislocation dynamics study, Int. J. Plasticity, 97 (2017) 1 – 23. [75] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, Second Edition, Chapman & Hall, London, 1997. [76] https://www.fiz-karlsruhe.de/de/leistungen/kristallographie/icsd.html, Feb. 2017, ICSD Database.

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[77] A. Ma, D. Dye, R.C. Reed, A model for the creep deformation behaviour of the superalloy single-crystal superalloy CMSX-4, Acta Mater. 56 (2008) 1657-1670.

M AN U

[78] S. Karthikeyan, R.R. Unocic, P.M. Sarosi, G.B. Viswanathan, D.D. Whitis, M.J. Mills, Modeling microtwinning during creep in Ni-based superalloys, Scripta Mater. 54 (2006) 1157-1162.

AC C

EP

TE D

[79] Y.M. Eggeler, M.S. Titus, A. Suzuki, T.M. Pollock, Creep deformation-induced antiphase boundaries in L12-containing single-crystal cobalt-base superalloys, Acta Mater. 77 (2014) 352-359.

26

ACCEPTED MANUSCRIPT

Tables

Table 1: g-vectors and the experimentally established deviation parameters w. g-vectors marked with “A” were used for 3D anaglyphs. An “F” indicates that the corresponding TEM

g

w



g A,F

g9

0.33

  A g3 : 202  A g4 : 202  g5: 2 00 F g6 : 1 1 1 g7: 0 2 2 A g8 :  1 1 1 A g2 : 1 1 1

0.50 0.88 1.03

SC

0.67 0.42

0.55



g10: 1 3 1

0.32

0.78



: 1 11

  F g11 : 2 20 g12: 3 11 A,F g13 : 1 1 1  A g14 : 1 1 1  g15:  1 1 3  F g16 : 002 

0.71

0.94

w

1.14 0.60

M AN U



A g1 : 1 1 1

RI PT

micrograph is shown in the present work.

0.50 1.37 0.80

-1

-4

1.9·10

3

-19

4.65·10

-29

1.09·10

[75]

T/K

[76]

Ac * -8

Ag *

Asc *

Asg *

1023

1.2·10

10·Ac

2·Ac

1.5·Ag

this work

this work

this work

this work

this work

EP

[75]

/m 

Q/J

TE

2

D0 / m s

D

Table 2: Input parameters used in the present work to describe diffusion controlled processes.

AC C

* dimension: m s-1 / (N m-1)

Table 3: A second set of material parameters and loading conditions used in the present work. a – lattice constant, μ – shear modulus,  - Poisson ration, S – applied stress, M – misfit stress, ESF – stacking fault energy, ESISF – super lattice intrinsic stacking fault energy and EAPB – antiphase boundary energy.

-10

3.52·10 [76]

ESF

ESISF

EAPB



S / Nm

74.8·10

0.3

8·10

4·10

30

12

170

[75]

[75]

this work

[13]

[38]

[54]

[18]

μ / Nm

a/m

-2

9

-2

M / Nm

8

8

1

-2

mJm

-2

mJm

-2

mJm

-2

ACCEPTED MANUSCRIPT

g

d1

d2

d3

d4

d5

+

±1

+

±1

+

±1

+

±1

+

3,4

+

±1

+

±1

+

±1

+

±1

+

5

-

0

-

0

-

0

-

0

-

6

+

±1

+

±1

+

±1

+

±1

7

+

±2

+

±2

+

±2

+

±2

8,9

+

±1

+

±1

+

10

+

±2

+

±2

+

11

+

±1

+

±1

+

12

res

0

res

0

res

13,14

-

0

-

0

-

15

+

±2

+

±2

16

+

±1

+

±1





 1/2 01 1



±1

+

±1

+

±1

±1

+

±1

+

±1

0

-

0

-

0

±1

+

±1

+

±1

+

±2

+

±2

+

±2

M AN U

+

+

±1

+

±1

+

±1

+

±1

±2

+

±2

+

±2

+

±2

+

±2

±1

+

±1

+

±1

+

±1

+

±1

0

res

0

res

0

res

0

res

0

0

-

0

-

0

-

0

-

0

+

±2

+

±2

+

±2

+

±2

+

±2

+

±1

+

±1

+

±1

+

±1

+

±1

D

±1

TE



 1/2 01 1

d7



 1/2 01 1





 1/2 01 1





 1/2 01 1





 1/2 01 1





 1/2 01 1



AC C

EP

b

d6

SC

1,2

RI PT

Table 4a: Effective visibility (visible: +, invisible: -, residual contrast: res) for dislocations 1 to 7 as identified in Figure 1 together with the scalar product between contrast condition g and Burgers vector b. Left column: g-vectors as listed in Table 1. All other columns: visibilities and scalar products of dislocations. For details see text.

Table 4b: Effective visibility (visible: +, invisible: -, residual contrast: res) for dislocations 8 to 14 as identified in Figure 1 together with the scalar product between contrast condition g and Burgers vector b. Left column: g-vectors as listed in Table 1. All other columns: visibilities and scalar products of dislocations. For details see text. 2

ACCEPTED MANUSCRIPT

g

d8

d9

d 10

d 11

d 12

d 13

d 14

+

±1

+

±1

+

+2/3

-

0

-

0

-

-1/3

+

-2/3

3,4

+

±1

+

±1

+

+2

+

±2

+

±2

+

-1

+

-2

5

-

0

-

0

res

-2/3

+

±1

+

6

+

±1

+

±1

res

-2/3

res

0

res

7

+

±2

+

±2

+

-2

+

±1

+

8,9

+

±1

+

±1

+

-4/3

+

±1

+

10

+

±2

+

±2

+

-4/3

-

0

11

+

±1

+

±1

-

0

+

±1

12

res

0

res

0

+

-4/3

+

±2

13,14

-

0

-

0

+

+2/3

+

15

+

±2

+

±2

+

-8/3

16

+

±1

+

±1

res

+4/3

b

 1/2 01 1



 1/2 01 1



-

+1/3

+

+2/3

0

-

+1/3

+

+2/3

±1

+

+1

+

+2

±1

+

+2/3

res

+4/3

-

0

res

-2/3

+

+4/3

+

±1

-

0

-

0

+

±2

+

+2/3

res

+4/3

±1

+

±1

-

-1/3

+

-2/3

+

±2

+

±2

+

+4/3

+

+8/3

+

±1

+

±1

+

-2/3

+

-4/3

SC

±1

M AN U



 



 1/2 10 1

1/3 112

AC C

EP

TE

D



RI PT

1,2

3





 1/2 10 1







1/6 1 1 2





1/3 1 1 2

EP TE D

M AN US C

ED

M AN

TE D

M AN U

ED

M AN

AC C

EP

TE D M AN US C

RI

CC

EP TE D

M AN US C

RI

ED M AN

D

M AN

CE PT ED

M AN US C

R

D

M AN