Crystal structure and composition dependence of mechanical properties of single-crystalline NbCo2 Laves phase

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Crystal structure and composition dependence of mechanical properties of single-crystalline NbCo2 Laves phase W. Luo , C. Kirchlechner , J. Zavaˇsnik , W. Lu , G. Dehm , F. Stein PII: DOI: Reference:

S1359-6454(19)30767-0 https://doi.org/10.1016/j.actamat.2019.11.036 AM 15662

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Acta Materialia

Received date: Revised date: Accepted date:

23 May 2019 12 November 2019 13 November 2019

Please cite this article as: W. Luo , C. Kirchlechner , J. Zavaˇsnik , W. Lu , G. Dehm , F. Stein , Crystal structure and composition dependence of mechanical properties of single-crystalline NbCo2 Laves phase, Acta Materialia (2019), doi: https://doi.org/10.1016/j.actamat.2019.11.036

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Title Crystal structure and composition dependence of mechanical properties of single-crystalline NbCo2 Laves phase

Authors W. Luo a,*, C. Kirchlechner a, J. Zavašnik a,b, W. Lu a, G. Dehm a and F. Stein a,*

Affiliation address a

Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße 1, D-40237 Düsseldorf,

Germany b

Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia

*Corresponding author Wei Luo

email: [email protected]; [email protected]

Frank Stein

email: [email protected]

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Abstract Extended diffusion layers of the cubic C15 and hexagonal C14 and C36 NbCo 2 Laves phases with concentration gradients covering their entire homogeneity ranges were produced by the diffusion couple technique. Single-phase and single-crystalline micropillars of the cubic and hexagonal NbCo2 Laves phases were prepared in the diffusion layers by focused ion beam (FIB) milling. The influence of chemical composition, structure type, orientation and pillar size on the deformation behavior and the critical resolved shear stress (CRSS) was studied by micropillar compression tests. The pillar orientation influences the activated slip systems, but the deformation behavior and the CRSS are independent of orientation. The deformation of the smallest NbCo2 micropillars (0.8 μm in top diameter) appears to be dislocation nucleation controlled and the CRSS approaches the theoretical shear stress for dislocation nucleation. The CRSS of the 0.8 μm-sized NbCo2 micropillars is nearly constant from 26 to 34 at.% Nb where the C15 structure is stable. It decreases as the composition approaches the Co-rich and Nb-rich boundaries of the homogeneity range where the C15 structure transforms to the C36 and the C14 structure, respectively. The decrease in the CRSS at these compositions is related to the reduction of shear modulus and stacking fault energy. As the pillar size increases, stochastic deformation behavior and large scatter in the CRSS values occur and obscure the composition effect on the CRSS.

Keywords: Laves phases; chemical diffusion; plasticity; micromechanics; composition effect

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1 Introduction Transition metal Laves phases are promising high temperature structural materials due to their high melting points, high strength and good creep resistance [1]. However, their applications are limited by the pronounced brittleness up to high homologous temperatures [2]. Their high strength and distinct brittleness originate mainly in their tightly packed and complex crystal structures where dislocations move with substantial difficulty [1]. The synchroshear mechanism, which requires the simultaneous motion of two Shockley partial dislocations on the adjacent planes of a triple layer, was proposed to explain the dislocation motion in Laves phases [3]. The deformation of Laves phases is a thermally activated process, and thus plasticity is usually observed at an elevated temperature where the elastic constants decrease and thermal activation is significant [2, 4]. Laves phases are thermodynamically stable in three different structure types, hexagonal C14 (MgZn2-type), cubic C15 (MgCu2-type) and hexagonal C36 (MgNi2-type) [5]. In many binary systems, especially transition metal systems, different Laves phase polytypes exist as equilibrium phases. Due to the close relation and small difference in the structural energy of different Laves phase polytypes, their stability sensitively depends on temperature, composition and pressure [6], and a small shift in temperature or composition may lead to a change of the stable structure type. However, little information about the influence of structure type of Laves phases on mechanical properties is reported in literature. Some binary Laves phases such as NbCo 2 and NbFe2 exhibit widely extended homogeneity ranges on both sides of the stoichiometric composition. The exact reasons for the wide homogeneity ranges and the stability of these Laves phases are not fully understood yet [5]. It has been reported that the mechanical properties of Laves phases can be influenced by their

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composition [7]. However, the effect of composition on the yield stress of Laves phases is still controversial and the underlying mechanism is not yet clear. Müller and Paufler [8] reported that the yield stress of the C14 MgZn2 Laves phase is maximum at the stoichiometric composition and drops as the composition deviates from the stoichiometry, which indicates softening at offstoichiometric compositions. However, Takasugi et al. [9] reported that there is no significant influence of composition on the yield stress of the C15 NbCr2 Laves phase. Moreover, Voβ et al. [10] showed that the compressive strengths of both the C14 NbFe2 and C15 NbCo2 Laves phases increase with increasing Nb content, which indicates softening on the Fe-/Co-rich side and hardening on the Nb-rich side. The aim of this work is to study the influence of crystal structure and chemical composition on the strength of transition metal Laves phases. The Co-Nb system [11] contains three Laves phase polytypes, C14, C15 and C36 NbCo2. The stable structure type changes from the C36 variant via C15 to C14 with increasing Nb content. In addition, the cubic C15 NbCo 2 Laves phase has an extended homogeneity range from approximately 26.0 to 35.3 at.% Nb [11]. These features allow a simultaneous study of the dependence of mechanical properties of the NbCo 2 Laves phase on composition and crystal structure. However, the brittleness of Laves phases makes it difficult and time-consuming to prepare flaw-less bulk Laves phase samples [12], and the high brittle-to-ductile transition temperature (BDTT) of transition metal Laves phases [2] makes the study of plasticity of bulk Laves phase samples challenging. In order to circumvent the difficulties in preparing and testing bulk Laves phase samples, a combination of the diffusion couple technique and micromechanical testing is adopted in the present work. The beauty of the diffusion couple technique is that diffusion layers of the three NbCo 2 Laves phases with concentration gradients covering their entire homogeneity ranges can be produced with only two

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diffusion couples. Micropillars of the cubic and hexagonal NbCo2 Laves phases with different sizes, orientations and compositions can be prepared in the diffusion layers by FIB milling. Micropillar compression has been demonstrated to be an effective method to study the plasticity of brittle materials even at room temperature [13-16].

2 Experimental details Two diffusion couples Co/Co-30Nb and Nb/Co-30Nb together covering the composition ranges of the three NbCo2 Laves phases were prepared. They are composed of pure Co, pure Nb and single-phase C15 NbCo2 Laves phase alloys with composition of 30 at.% Nb. While the Co/Co-30Nb diffusion couple was heat-treated at 1200 oC for 96 h, the Nb/Co-30Nb diffusion couple was heat-treated at 1350 oC for 24 h. The heat treatments were performed in an argon atmosphere. After heat treatment, the diffusion couples were furnace-cooled. The choice of the heat treatment temperatures is determined by the phase diagram [11]. The C36 Laves phase is only stable between about 1040 and 1264 °C and there is a deep lying eutectic at 1239 °C in the Co-rich part of the system. Therefore, a heat treatment temperature of 1200 °C was chosen. The C14 Laves phase on the Nb-rich side exists between about 1250 and 1424 °C and there are two Nb-rich eutectics at 1379 and 1364 °C. Here, 1350 °C was selected as heat treatment temperature. The microstructures of the diffusion couples were observed by optical microscopy (Carl Zeiss light optical microscope) and scanning electron microscope (SEM, JSM 6490, JEOL). The chemical concentration profiles of the diffusion couples were measured by electron probe microanalyzer (EPMA) using wavelength dispersive X-ray spectroscopy (WDS) with a step width of 1 µm. Three scans were measured for each diffusion couple. EPMA measurements were carried out with a camera SX 50 and a JEOL JXA 8100 instrument using pure Co and Nb as standards. 5

The crystallographic grain orientations were determined by EBSD using an EDAX/TSL detector setup with a Hikari CCD camera and a JEOL JSM 6490 SEM. Cylindrical micropillars of the cubic and hexagonal NbCo 2 Laves phases with 0.8 µm, 3 µm and 5 µm in top diameter and an aspect ratio of about 3 were cut in the corresponding diffusion zones by FIB milling using a Zeiss Auriga dual beam FIB/SEM instrument. As there are concentration gradients in the extended diffusion zones, micropillars with different compositions can be obtained by FIB milling at different distances with respect to the interfaces. Their compositions can be estimated from the concentration profiles and their orientations were determined by EBSD measurements. Ga+ beam currents of 16 nA and 4 nA at 30 kV were used for coarse milling and the final milling was carried out using 240 pA or 120 pA at 30 kV depending on the pillar size. The resulting taper angle of the pillars is around 2 ‒ 3 o. Micropillars of the C15 NbCo2 Laves phase with single-slip orientation ( 123 or 135 ) and multiple-slip orientation 100 were prepared to study the effect of orientation on the deformation behavior. The Schmid factors for the

1 1 110 -type full and 112 -type partial 2 6

dislocations associated with the primary slip system of the C15 NbCo 2 micropillars with [100] ,

[123] and [135] orientations are listed in Table 1. In order to study the effect of orientation on the deformation behavior of the hexagonal C14 and C36 NbCo 2 Laves phases, micropillars oriented for basal and non-basal slip were prepared. As shown in Table 2, the orientations of the hexagonal C14 and C36 NbCo 2 micropillars are denoted by the inclination angle between the loading axis and the basal plane. As different inclination angles of the loading axis to the basal plane lead to different Schmid factors for various slip systems, micropillars with specific orientations of nearly 0o, 40o and 90o inclined to the (0001) plane were prepared. While basal slip is expected when the inclination angle is about 40o, prismatic and pyramidal slip are expected 6

when the inclination angles are 0o and 90o, respectively. Similar to the dislocation slip on the

{111} planes of the C15 NbCo2 Laves phase, the relative Schmid factors for the two associated 1 1100 -type partial dislocations may influence the basal slip of the hexagonal NbCo 2 Laves 3 phases. Therefore, the C14 and C36 NbCo 2 micropillars oriented for basal slip were further divided into two groups. One group is oriented for (0001)1120 slip and the other group is oriented for (0001)1100 slip. The orientations and Schmid factors for the basal and non-basal slip of the hexagonal C14 and C36 NbCo 2 micropillars are listed in Table 2. The micropillar compression tests were conducted using an ASMEC UNAT‒2 indenter installed in a JEOL JSM 6490 SEM or a Zeiss Gemini 500 SEM. Diamond flat punch indenters (Synton-MDP, Switzerland) were used to load the micropillars in displacement-controlled mode at a strain rate of 10 -3/s. In situ videos of the compression tests and the corresponding forcedisplacement curves were recorded. Slip trace analysis is based on the pillar orientation obtained from EBSD measurements, the in situ videos and post mortem SEM images. As the plastic deformation of the micropillars usually starts at the top surface, which has the smallest crosssectional area due to tapering, the stress was calculated using the applied load and the area of the top surface. The engineering strain was calculated using the recorded displacement data of the indenter and the initial pillar height. Neither the sink-in nor the instrument frame compliance was corrected in the load-displacement data. For the uncorrected data it is clear that larger micropillars (sustaining higher loads) show a more prominent effect of the instrument frame compliance than smaller micropillars, leading to different apparent loading stiffness of the micropillars with different sizes. In our study, a quantitative assessment of effects of specimen geometry and base materials on the elastic behavior of micropillars as performed by Yang et al.

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in [17] is not attempted. Instead, the indentation modulus of the NbCo 2 Laves phase was studied using nanoindentation [18]. As it is an independent research exceeding the limits of the current paper, the results will be published separately. Furthermore, a dual beam FIB/SEM instrument (Helio Nanolab 600i, FEI) was used to prepare transmission electron microscopes (TEM) samples from a deformed micropillar. Details of the deformation mechanism of the micropillar were analyzed by image-corrected TEM and probecorrected scanning TEM (S/TEMs, Titan Themis 60–300, FEI). Both instruments were operated at 300 keV and are equipped with a high-brightness field emission gun. In the case of the probe corrected STEM, a Fischione Instruments (Model 3000) high angle annular dark field detector (HAADF) was used for STEM imaging.

3 Results 3.1 Microstructure and concentration profiles of the diffusion couples The microstructures and the corresponding concentration profiles of the Co/Co-30Nb and Nb/Co-30Nb diffusion couples after annealing are shown in Fig. 1(a) and (b), respectively. In the Co/Co-30Nb diffusion couple, the thickness of the diffusion zones of C36 and C15 NbCo 2 is about 70 µm and 400 µm, respectively. A diffusion layer of C14 NbCo 2 and an extended diffusion zone of C15 NbCo 2 exist in the Nb/Co-30Nb diffusion couple and their thickness is 40 and 200 µm, respectively. The diffusion layers of the C14 and C36 NbCo 2 Laves phases consist of coarse columnar grains with tens of micrometers in size. The grain size of the C15 NbCo 2 Laves phase is roughly 100 µm. The homogeneity ranges of the C14, C15 and C36 NbCo2 Laves phases measured from the concentration profiles of the two diffusion couples are summarized in Table 3. The error bars of the compositions are determined from the measured concentration

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profiles. The homogeneity ranges of the C36 and C14 NbCo2 Laves phases obtained from the diffusion couples are from 24.0 ± 0.5 to 25.0 ± 0.5 at.% Nb and from 35.5 ± 0.5 to 37.0 ± 0.5 at.% Nb, respectively. While the two hexagonal NbCo 2 Laves phases exist in narrow composition ranges, the cubic C15 NbCo 2 Laves phase is stable in a large composition range from 25.0 ± 0.5 to 34.3 ± 0.5 at.% Nb.

3.2 Analysis of slip traces and stress-strain curves 3.2.1 Cubic C15 NbCo2 After compression, the 0.8 µm-sized pillars of C15 NbCo 2 (Fig. 2(a)) with single-slip orientation exhibit large slip steps on the surface and complex morphologies caused by severe deformation. The large slip steps observed in the side view are identified as traces of the {111} planes. However, the slip direction cannot be clearly determined from the top view of a single deformed micropillar due to the severe deformation of the pillar top. A series of 0.8 µm pillars with the same composition and orientation were compressed and their post mortem images suggest that the slip direction is 110 -type. Their engineering stress-strain curves (Fig. 2(f)) show elastic deformation followed by a large strain burst. The large strain burst and the activation of the {111}110 slip systems observed in the 0.8 µm-sized C15 NbCo2 micropillars are in agreement with the results reported in [19]. Similarly, most of the 3 µm-sized C15 NbCo2 micropillars with single-slip orientation (Fig. 2(b)) show large slip steps emanating from the pillar top and crossing the entire pillar, indicating the activation of the {111}110 slip systems. A large strain burst at the onset of yielding was also observed in their stress-strain curves (Fig. 2(f)). However, a few large micropillars of C15 NbCo 2 with 5 µm in top diameter (Fig. 2(c)) show fine slip traces on the surfaces instead of cracks and complex morphologies resulting from 9

the plastic deformation bursts. Their corresponding stress-strain curves (Fig. 2(f)) exhibit elastic deformation followed by intermittent small strain bursts at lower stresses. In order to analyze the slip traces, loading was stopped after the first small strain burst to avoid severe deformation. The slip traces of the 5 µm-sized C15 NbCo2 micropillars with single-slip orientation are identified to be the {111}110 slip systems. The deformation behavior of the 0.8 µm-sized C15 NbCo2 micropillars with 100 orientation (Fig. 2(d)) is similar to that of the single-slip oriented C15 NbCo2 micropillars, but the slip traces indicate the activation of the {111}112 slip systems in the 100 oriented micropillars. In contrast to the 0.8 µm-sized micropillars which have experienced catastrophic failure, some 3 µm-sized C15 NbCo2 micropillars with 100 orientation show elastic deformation followed by small strain bursts at much lower stresses (Fig. 2(f)), creating fine slip traces of the {111}112 slip systems located in the middle and the bottom of the pillars (Fig. 2(e)). The maximum stress reached prior to the first strain burst is considered as the yield stress. The average yield stresses of the 0.8 µm-sized C15 NbCo2 micropillars with single-slip and 100 orientations at 30 at.% Nb are 10 ± 1 and 9.6 ± 0.8 GPa, respectively. For the 3 µm-sized micropillars, the respective values amount to 8.6 ± 0.5 and 8 ± 1 GPa. The errors of the average yield stresses represent the standard deviations. While the deformation behavior becomes stochastic and the yield stress decreases as the pillar diameter increases, the orientation does not show any significant influence on the deformation behavior and the yield stress of the C15 NbCo2 micropillars.

3.2.2 Hexagonal C36 NbCo2

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The 0.8 µm- and 3 µm-sized 36o (0001) micropillars of C36 NbCo 2 are oriented for

(0001)1120 slip. As shown in Fig. 3(a) and 3(b), they both exhibit large slip steps at the pillar top. The identified slip traces and slip direction confirm the activation of the (0001)1120 slip systems in these micropillars. A large strain burst at the onset of yielding was observed in their stress-strain curves (Fig. 3(g)). Similar deformation behavior was observed in the 0.8 μm- (Fig. 3(c)) and 3 μm-sized (Fig. 3(d)) 39o (0001) micropillars of C36 NbCo 2 which are oriented for

(0001)1100 slip. The 3 μm-sized C36 NbCo2 micropillars with 84o (0001) (Fig. 3(e)) and 9o (0001) (Fig. 3(f)) orientations were prepared for observation of non-basal plane slip due to the low Schmid factor for basal slip. While the slip traces of the 84 o (0001) micropillars of C36 NbCo2 indicate the activation of the {1012}1011 -type pyramidal slip, the {1100}1120 -type prismatic slip is identified to operate in the 9o (0001) micropillars of C36 NbCo 2. As shown in Fig. 3(g), their corresponding stress-strain curves show elastic deformation followed by a limited amount of plastic deformation or a large strain burst. The yield stresses of the 0.8 µm-sized C36 NbCo2 micropillars oriented for (0001)1120 and

(0001)1100 slip are 9 ± 1 and 9.9* GPa, respectively, and the respective values of the 3 µmsized C36 NbCo2 micropillars are 6 ± 2 GPa and 6.5 ± 0.9 GPa. The yield stresses of the 3 μmsized C36 NbCo2 pillars with 9o (0001) and 84o (0001) orientations, which are designed for prismatic and pyramidal slip, are 5.7 ± 0.2 and 5.8 ± 0.4 GPa, respectively.

3.2.3 Hexagonal C14 NbCo2

*

Only one 0.8 μm-sized micropillar of C36 NbCo2 with 39o (0001) orientation was tested.

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The slip traces of the 0.8 µm- (Fig. 4(a)) and 3 µm-sized (Fig. 4(b)) C14 NbCo2 micropillars with 37o (0001) orientation indicate the activation of the (0001)1100 slip systems. The 3 µmsized 35o (0001) C14 NbCo2 micropillars (Fig. 4(c)), which are oriented for (0001)1120 slip, were severely deformed and show complex morphologies consisting of cracks and multiple slip traces. Repeated compression tests of a series of C14 NbCo 2 micropillars having the same orientation indicate the activation of the (0001)1120 slip systems. Based on the slip trace analysis of a series of deformed pillars, {1011}1012 -type pyramidal slip and {1100}1120 -type prismatic slip are identified to operate in the 69o (0001) (Fig. 4(d)) and 6o (0001) (Fig. 4(e)) oriented C14 NbCo2 micropillars, respectively. In Fig. 4(f), all the stress-strain curves of the C14 NbCo 2 micropillars oriented for basal and non-basal slip show elastic deformation followed by a large strain burst. While the yield stresses of the 0.8 µm-sized C14 NbCo2 micropillars oriented for (0001)1100 slip is 9 ± 1 GPa, the yield stresses of the 3 µm-sized C14 NbCo2 micropillars oriented for (0001)1120 and

(0001)1100 slip are 5.6 ± 0.2 and 6.4 ± 0.6 GPa, respectively. The yield stresses of the 3 µmsized C14 NbCo2 micropillars oriented for prismatic and pyramidal slip are 5.3 ± 0.4 and 5 ± 1 GPa, respectively.

3.4 TEM analysis of a deformed C15 NbCo2 micropillar The activation of the {111}112 slip systems in the 100 oriented C15 NbCo2 micropillars indicates a partial dislocation-mediated deformation. This might lead to the formation of twins or martensitic phase transformation [20]. In order to verify the deformation mechanism, a TEM

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lamella was prepared from a deformed 3 µm-sized C15 NbCo2 micropillar with [010] † orientation. The sample consisting of a single crystal was tilted in [101] zone axis. The parallel dark lines observed in the TEM micrograph (Fig. 5(a)) correspond to the (111)[121] slip traces observed in the post mortem SEM image (inset). The corresponding selected area diffraction patterns (SADP) taken from the areas marked with red circles in Fig. 5(a) are shown in Fig 5(be). The SADPs taken from a bulk area or recorded over the planar defect areas are identical and no evidence for twinning was found. In Fig. 6(a), the HAADF STEM image viewed along [101] direction and recorded over the planar defect reveals that the lattice is shifted with respect to the lattice below, which indicates the planar defect is a stacking fault. The formation of the stacking fault can be well explained by the synchroshear mechanism [3]. Fig. 6(b) and 6(c) schematically show the [101] projection of the C15 NbCo2 Laves phase before and after deformation, respectively. The stacking sequence of the C15 structure shown in Fig. 6(b) can be described as Aαcβ(X)-Bβaγ(Y)-Cγbα(Z)-Aαcβ(X)Bβaγ(Y)-Cγbα(Z). A

1 [121] synchro-Shockley partial dislocation, which consists of a pair of 6

Shockley partial dislocations lying on the adjacent planes of a triple layer γbα, glides through the

(111) plane. The triple layer γbα is changed to its twin-related variant γaβ. Consequently, the stacking sequence of the C15 structure becomes Aαcβ(X)-Bβaγ(Y)-Cγǀaβ(Z’)-Bβaγ(Y)-Cγbα(Z)Aαcβ(X), where the vertical line (in the middle of Z’) denotes the stacking fault left by the glide of the partial dislocation.

3.5 The critical resolved shear stress (CRSS) †

Note that specific crystallographic directions are used for the TEM analysis.

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The maximum stress prior to the first strain burst is taken as the yield stress. In the case of the C15 NbCo2 micropillars with single-slip orientation, the highest Schmid factor for the primary

{111}110 slip system is taken into account in the calculation of the CRSS. For the 100 oriented C15 NbCo2 micropillars, the highest Schmid factor for the primary {111}112 slip system is used to calculate the CRSS. Similarly, the highest Schmid factors for the primary

(0001)1120 and (0001)1100 slip systems are used to calculated the CRSS values of the hexagonal C14 and C36 NbCo 2 micropillars oriented for basal slip along 1120 and 1100 directions, respectively. The CRSS of non-basal slip is calculated based on the identified slip systems. The CRSS τ is calculated from the yield stress σ and Schmid factor m for the corresponding activated slip system by Schmid’s law:

 = m  .

(1)

The CRSS of the 0.8 µm-sized micropillars of the cubic and hexagonal NbCo 2 Laves phases is very high and close to the theoretical shear strength µ/30 ‒ µ/2π [21] where µ is the shear modulus. The average CRSS decreases as the pillar size increases. In addition, the larger pillars show a larger scatter in the CRSS values. The standard deviation of the CRSS is in the range of 20 % of the mean value for the 3 µm-sized pillars whereas it is about 10 % of the mean value for the 0.8 µm-sized pillars.

4 Discussion 4.1 The effect of pillar size Most of the NbCo 2 micropillars, especially the small 0.8 μm-sized micropillars, show elastic deformation followed by a large strain burst in the stress-strain curves and they exhibit complex morphologies resulting from the severe deformation during the strain burst regardless of 14

composition and crystal structure. The catastrophic failure does not represent the materials behavior but most likely is related to the intrinsic compliance of the non-ideally displacementcontrolled indenter after slip initiation. As the current displacement-controlled machines are actually ―spring-controlled machines‖, sudden load drops will result in a mix of load drop and displacement burst, where the magnitude of both strongly depends on the spring stiffness of the machine [22]. The inability of the indenter to maintain control during the strain softening leads to displacement bursts and extra deformation of the micropillars at very high strain rates, which may cause severe deformation and even failure of the micropillars. The stress where the first displacement burst occurs can be used to evaluate the yield stress, but the subsequent strain softening cannot be measured. Similar behavior was also reported in testing strain-softening materials which undergo dislocation nucleation and rapid multiplication using a load-controlled testing apparatus [23, 24]. Considering the fact that the dislocation density in the cubic and hexagonal NbCo 2 Laves phases in the as-cast state is low (in the order of 1011/m2) [18], it is expected that there are few dislocations in the micropillars, especially in the smallest ones with only 0.8 µm in top diameter. The very high CRSS and large strain burst observed in the 0.8 µm-sized NbCo2 micropillars could be related to the dislocation nucleation and the subsequent dislocation avalanches. Some large micropillars with 3 µm or 5 µm in top diameter may contain pre-existing mobile dislocations or dislocation sources, and thus they yield at substantially lower stresses and exhibit plastic deformation with intermittent small strain bursts. The slip traces usually start at the positions where the activated pre-existing dislocation sources are located. Generally, as the pillar size increases, the probability to find mobile dislocations or dislocation sources in a micropillar increases, and thus stochastic deformation behavior and larger scatter in the CRSS occur [24].

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The CRSS as a function of the pillar diameter of the C14, C15 and C36 NbCo 2 Laves phase is plotted in Fig. 7. The data were fitted with a non-linear model by the Mathematica® program applying the approach of Dou and Derby [25] following the equation

( /  )  A( D / b)n

(2)

where b is the length of the Burgers vector, D is the pillar diameter, and A and n are fitting constants. In Fig. 7, the CRSS values measured from the micropillars of the cubic and hexagonal NbCo2 Laves phase with various sizes are presented. In addition, the 95 % confidence intervals for the data of the three Laves phases are indicated as dashed lines. Based on the lattice parameters reported in [11], the Burgers vectors of a Shockley partial dislocation in C14, C15 and C36 NbCo 2 are calculated to be 0.2790, 0.2756 and 0.2739 nm, respectively. The shear modulus, which is calculated from the indentation modulus measured by nanoindentation, is 101.6, 111.5 and 106.5 GPa for C14, C15 and C36 NbCo 2, respectively [18]. The values of the exponent n in eq. (2) for C14, C15 and C36 NbCo 2 are -0.318 ± 0.04, -0.345 ± 0.06 and -0.296 ± 0.07, respectively. The magnitudes of these n values are significantly lower than that obtained for pure fcc metals (where n = -0.66) [25], indicating that the size effect on the yield stress of the cubic and hexagonal NbCo 2 Laves phases is less pronounced. This is presumably due to the low dislocation density and high Peierls stress, as discussed in [15, 26, 27]. Additionally, the size dependence of the CRSS is nearly identical for the three NbCo 2 Laves phases with different structure types.

4.2 The effect of crystallographic orientation 4.2.1 {111}/basal plane slip

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The deformation of the micropillars of the cubic and hexagonal NbCo 2 Laves phases, especially the 0.8 μm-sized micropillars, is found to be controlled by the nucleation and propagation of partial dislocations. It has been confirmed in MD simulations [28-30] that the relative Schmid factors for the leading and trailing partial dislocations can influence the nucleation of partial dislocations and result in different deformation mechanisms. In orientations where the Schmid factor for the trailing partial is higher than the corresponding leading partial, the trailing partial nucleates immediately after the emission of the leading partial and follows it closely, leading to a full dislocation slip. In orientations where the Schmid factor for the leading partial is higher than for the trailing partial, only the leading partial nucleates and glides, creating stacking faults or twins. Fig. 8 shows a schematic of the influence of orientation on the deformation behavior of the NbCo2 micropillars under compression. mL and mT are the Schmid factors for the leading and trailing partial dislocation, respectively. For the single-slip ( 123 or 135 ) oriented micropillars of C15 NbCo 2 under compression (Fig. 8(a)), the Schmid factor for the trailing partial is higher than the leading partial as shown in Table 1. In Fig. 8(b), the trailing partial can be almost simultaneously emitted with the leading partial and the pair of partial dislocations acts as a full dislocation. Thus, the {111}110 slip systems are activated. For compression of the

100 oriented C15 NbCo2 micropillars (Fig. 8(c)), the Schmid factor for the leading partial is higher than the trailing partial. As shown in Fig. 8(d), the leading partial nucleates at the surface and glides in a 112 -type direction on the {111} plane without emission of the trailing partial, which creates a widely extended stacking fault. Basal slip is the main deformation mechanism of the hexagonal C14 and C36 NbCo 2 micropillars with roughly 40o (0001) orientation. The basal slip direction can also be influenced 17

by the relative Schmid factors for the two partial dislocations. When a C14 or C36 NbCo2 micropillar is in an orientation where the two partial dislocations associated with the primary

(0001)1120 slip system have similar Schmid factors, both the two partials can nucleate and glide under the applied stress, and thus the overall slip is along a 1120 -type direction. When an associated partial dislocation has a significantly higher Schmid factor than the other one, it is exposed to a larger shear stress, and thus it can nucleate and glide along a 1100 -type direction leaving the other partial behind. Therefore, the (0001)1100 slip systems are activated. For both the cubic and hexagonal NbCo 2 Laves phases, the relative Schmid factors for the two associated partial dislocations influence the dislocation slip on their close-packed planes ({111} for C15 and (0001) for C14 and C36). In Table 4, a comparison of the CRSS of the cubic and hexagonal NbCo2 micropillars oriented for full dislocation slip and partial dislocation slip on the close-packed planes reveals that the CRSS values for the two sets of orientations are nearly identical. This indicates that, although the pillar orientation can influence activated slip systems, the energy barriers of dislocation nucleation and motion, which determine the strength of the NbCo2 micropillars, is independent of orientation.

4.2.2 Non-basal plane slip Non-basal plane slip systems including pyramidal slip ( {1011}1012 for C14 and {1012}1011 for C36) and prismatic slip {1100}1120 were observed in the micropillars of the hexagonal C14 and C36 NbCo2 Laves phases. Basal plane slip of the hexagonal Laves phases has been confirmed by several researchers [31, 32] and it can be explained by the operation of the synchroshear mechanism [3]. However, only little information of the non-basal plane slip in the

18

hexagonal Laves phases was reported in literature [33-35] and the mechanisms are not yet clear. Therefore, the CRSS of the non-basal slip can only be calculated based on the identified slip systems. As shown in Table 5, in contrast to hcp metals such as Mg, the CRSS values for basal and non-basal slip of the hexagonal C14 and C36 NbCo 2 Laves phases are both very high and the differences are within the error bars. Given the temperature and the strain rate are constant, the slip system which has the highest Schmid factor is most likely to be activated in the hexagonal C14 and C36 NbCo2 Laves phases. The activation of both basal and non-basal slip was also found in other hexagonal C14 [35] and C36 [33, 34] Laves phases after deformation at room temperature. Similar CRSS values for basal and non-basal slip of hexagonal C14 Mg2Ca was reported in [35].

4.3 The effect of chemical composition and crystal structure The activation of a partial dislocation source is energetically more favorable than the activation of a full dislocation source in Laves phases [36]. The energy E for the activation of a complete partial dislocation source is considered to be equal to the energy required for creating a complete partial dislocation loop [37]:

1 2  4R E   R 2 b  b 2 R( ) ln( 2 )   R 2 4 1  e

(3)

where τ is the resolved shear stress, b is the Burgers vector of partial dislocation, R is the dislocation loop radius, µ is the shear modulus, ν is the Poisson’s ratio, γ is the stacking fault energy, e is Euler’s number and ρ is the dislocation core cutoff which is assumed to be equal to the b 2 [37]. The first term is the work done by the applied shear stress, the second term is the self-energy of a partial dislocation loop, and the third term is the stacking fault energy of the

19

E 2 E corresponding partial dislocation loop. When the conditions  0 and  0 are achieved, R R 2 the corresponding stress is the theoretical shear stress for nucleation of a complete partial dislocation loop [38]. The following equation for the theoretical shear stress is obtained:

 theoretical 

 b 2   ( ) . 2 e2  1  b

(4)

The Poisson’s ratio ν of C15 NbCo2 is 0.28 [39] and it is assumed to be the same for all the cubic and hexagonal NbCo2 Laves phases. The stacking fault energy of the C15 NbCo 2 Laves phase at stoichiometric composition at room temperature is estimated to be 24 mJ/m2 [18]. The shear modulus μ of the C15 NbCo 2 Laves phase at the stoichiometric composition obtained from nanoindentation tests is 113 GPa [18], which is close to the calculated shear modulus of 114.4 GPa [39]. According to eq. (4), the calculated theoretical shear stress for nucleation of a partial dislocation loop at the stoichiometric composition is 11.7 GPa, which is close to μ/10. According to Bei et al. [38], an incomplete dislocation loop can be nucleated at a free surface or edge of a micropillar and the critical stress for dislocation nucleation of a micropillar is lower than the theoretical shear stress by a factor c

 pillar  c theoretical

(5)

where c is the correction factor that depends on the arc angle of the incomplete dislocation loop and the angle between the loading axis and the slip plane. For a half dislocation loop nucleated at a flat side surface or a quarter dislocation loop nucleated at an edge of a rectangular micropillar, c is estimated to be 0.5 and 0.3, respectively [38]. Since the arc angle of an incomplete dislocation loop nucleated at the surface of a circular micropillar (Fig. 9) is between that of a half loop and a quarter loop in a rectangular micropillar, c = 0.4 is a reasonable estimation for circular micropillars. Therefore, the estimated theoretical shear stress of a circular NbCo2 micropillar is 20

about μ/24, which agrees well with the CRSS measured from the 0.8 µm-sized NbCo2 micropillars. As shown in eq. (4) and (5), the theoretical shear stress of the NbCo 2 micropillars is a function of shear modulus, stacking fault energy and Burgers vector of partial dislocation. As the Nb content increases from 24 at.% to 37 at.%, the structure type of the NbCo 2 Laves phase changes from hexagonal C36 via cubic C15 to hexagonal C14. The value of b only increases by 1.8 % with increasing Nb, which indicates the effect of composition and crystal structure on the magnitude of Burgers vector is negligible. The electron channeling contrast imaging (ECCI) observations on the diffusion couples show that the cubic C15 NbCo 2 Laves phase exhibits only few individual dislocations in the composition range from about 26 to 34 at.% Nb but a high density of widely extended stacking faults at about 25 and 34.3 at.% Nb where the Co-rich and Nb-rich boundaries of C15 NbCo 2 are, respectively [18]. Widely extended stacking faults were also observed in the C14 and C36 NbCo 2 Laves phases [18]. It indicates the stacking fault energies at the phase boundaries of the C15 NbCo 2 Laves phase as well as the homogeneity ranges of the C14 and C36 NbCo 2 Laves phases are very low. Moreover, nanoindentation tests show that the indentation modulus of the NbCo2 Laves phase is nearly constant from 26 to 34 at.% Nb whereas it decreases as the composition approaches Co-rich and Nb-rich boundaries of the homogeneity range presumably due to the reduced phase stability at these compositions [18]. The dependence of the theoretical shear stress of circular micropillars on the stacking fault energy and shear modulus is shown in Fig. 9. The theoretical shear stress increases with increasing stacking fault energy and shear modulus. An increase of stacking fault energy from 0 mJ/m2 to 100 mJ/m2 only leads to an increase of 0.144 GPa in the theoretical shear stress, whereas the theoretical shear stress increases proportionally as the shear modulus increases.

21

Therefore, the influence of the shear modulus on the theoretical shear stress of the NbCo 2 micropillars is larger than that of the stacking fault energy. The experimental and theoretical CRSS values of the micropillars of the cubic and hexagonal NbCo2 Laves phases as a function of the composition are shown in Fig. 10. The shear modulus of NbCo2 as a function of composition obtained from nanoindentation tests [18] was used in the calculations of the theoretical CRSS and the standard deviation of the shear modulus was used to evaluate the uncertainty of the theoretical CRSS. The experimental CRSS values measured from the 0.8 µm-sized micropillars agree well with the calculated CRSS values, which indicates that the strength of the 0.8 µm-sized micropillars is mainly determined by dislocation nucleation at pillar surfaces. It also shows that the CRSS of 0.8 µm-sized micropillars is nearly constant from 26 to 34 at.% Nb where the C15 structure is stable. It decreases as the composition approaches the Co-rich and Nb-rich boundaries of the homogeneity range where the C15 structure transforms to the C36 and the C14 structure, respectively. This is consistent with the compositional trends of the stacking fault energy and the shear modulus. Therefore, the decrease in the CRSS of the 0.8 µm-sized micropillars of the NbCo 2 Laves phase at Co-rich and Nb-rich boundaries of the homogeneity range is mainly due to the reduction of shear modulus and stacking fault energy. As the pillar size increases, the probability to find pre-existing mobile dislocations in a micropillar is higher, leading to a lower CRSS value. Moreover, the variation in dislocation distribution of the large pillars leads to stochastic deformation behavior and larger scatter in the CRSS values, which could mask the composition effect.

5 Conclusions

22

In order to circumvent the difficulties in preparing and testing bulk brittle Laves phase alloys and to study the effect of chemical composition, structure type, pillar size and orientation on the deformation behavior and the CRSS of the NbCo 2 Laves phase in detail, a new approach, which is to combine the diffusion couple technique and micromechanical testing, is used in the present work. The main conclusions are as follows:

1. 0.8 µm-sized NbCo2 micropillars show elastic deformation followed by a large strain burst and their CRSS values are close to the theoretical shear stress, which indicates the deformation of the 0.8 µm-sized NbCo2 micropillars is dislocation nucleation controlled. The experimental CRSS of the 0.8 µm-sized NbCo2 micropillars is nearly constant from 26 to 34 at.% Nb where the C15 structure is stable. It decreases as the composition approaches the Co-rich and Nb-rich boundaries of the homogeneity range where the C15 structure transforms to the C36 and C14 structure, respectively. The decrease in CRSS at these compositions is primarily due to the reduction of the shear modulus and, to a lesser extent, the stacking fault energy. 2. As the pillar size increases, the probability to find pre-existing mobile dislocations in a micropillar is higher, and thus small intermittent strain bursts at lower stresses were observed in some large NbCo 2 micropillars with 3 µm and 5 µm in top diameter. The variation in dislocation distribution of the large micropillars leads to stochastic deformation behavior and larger scatter in the CRSS values, which may obscure the composition effect on the CRSS. 3. The observed dislocation slip on the close-packed planes ({111} for C15 and (0001) for C14 and C36) of the cubic and hexagonal NbCo 2 Laves phases either can be along

23

perfect dislocation slip directions ( 110 for C15 and 1120 for C14 and C36) or along partial dislocation slip directions ( 112 for C15 and 1100 for C14 and C36) depending on the relative Schmid factors for the two corresponding partial dislocations. 4. Both basal and non-basal slip were observed in the deformed micropillars of the hexagonal C14 and C36 NbCo 2 Laves phases. The CRSS values for basal and non-basal slip are comparable. Given the temperature and the strain rate are constant, the slip system which has the highest resolved shear stress is most likely to be activated.

24

Acknowledgements The authors thank Prof. K. Sharvan Kumar for fruitful discussions, Leon Christiansen and Angelika Bobrowski for their help in metallographic preparation and observations, as well as Nagamani Jaya Balila, Nataliya Malyar and Shunsuke Taniguchi for their help in preparation and compression of micropillars. Financial support from China Scholarship Council (CSC) (No. 201406370160) and International Max Planck Research School for Interface Controlled Materials for Energy Conversion (IMPRS-SurMat) is gratefully acknowledged.

25

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[39] S. Chen, Y. Sun, Y.-H. Duan, B. Huang, M.-J. Peng, Phase stability, structural and elastic properties of C15-type Laves transition-metal compounds MCo2 from first-principles calculations, J. Alloys Compd. 630 (2015) 202-208. Tables and table captions

Table 1 Schmid factors for the full and partial dislocations associated with the primary slip systems of the [100] , [123] and [135] oriented micropillars of C15 NbCo 2 under uniaxial compression. Compression axis Primary slip plane

Schmid factor Full dislocation Leading partial Trailing partial

1

[100] 1

(111) 1

1 1 1 0.41 ( [110] 1 ) 0.47 ( [211] 1) 0.24 ( [121] 1) 2 6 6

[123]

(111)

1 0.47 ( [101] ) 2

1 0.34 ( [112] ) 6

1 0.47 ( [211] ) 6

[135]

(111)

1 0.49 ( [101] ) 2

1 0.38 ( [112] ) 6

1 0.47 ( [211] ) 6

There are 8 slip systems with equivalent Schmid factor for the [100] oriented C15 NbCo 2

micropillars.

30

Table 2 Highest Schmid factors for basal and non-basal slip of the hexagonal C36 and C14 NbCo2 micropillars with various orientations under uniaxial compression. Orientation1

Phase

Schmid factor Basal slip

Basal slip

Prismatic slip

Pyramidal slip

Pyramidal slip

(0001)1120

(0001)1100

{1100}1120

{1011}1012

{1012}1011

C36

84o (0001)

0.11

0.09

0.01

―

0.46

C36

39o (0001)3

0.42

0.49

0.26

―

0.23

C36

36o (0001)2

0.47

0.41

0.29

―

0.29

C36

9o (0001)

0.15

0.14

0.47

―

0.41

C14

69o (0001)

0.33

0.31

0.06

0.49

―

C14

35o (0001)2

0.47

0.42

0.30

0.31

―

C14

37o (0001)3

0.45

0.48

0.31

0.37

―

C14

6o (0001)

0.10

0.09

0.49

0.42

―

1

Inclination of the loading axis relative to the basal plane

2

Oriented for (0001)1120 slip

3

Oriented for (0001)1100 slip

31

Table 3 Homogeneity ranges of the C36, C15 and C14 NbCo2 Laves phases measured from the diffusion couples by EPMA. Diffusion

Heat treatment

couple

Composition (at.% Nb) C36

C15

C14

Co/Co-30Nb

1200 oC, 96 h

24.0 ± 0.5 – 25.0 ± 0.5

25.0 ± 0.5 – 29.7 ± 0.5

―

Nb/Co-30Nb

1350 oC, 24 h

―

29.7 ± 0.5 – 34.3 ± 0.5

35.5 ± 0.5 – 37.0 ± 0.5

32

Table 4 The CRSS of the micropillars of the cubic and hexagonal NbCo 2 Laves phases oriented for full dislocation slip ( {111}110 for C15 and (0001)1120 for C36 and C14) and for partial dislocation slip ( {111}112 for C15 and (0001)1100 for C36 and C14). Phase

Pillar size [μm]

Orientation

C36 NbCo2

0.8

36o (0001)

(0001)1120

4.3 ± 0.7

C36 NbCo2

0.8

39o (0001)

(0001)1100

4.2*

C36 NbCo2

3

36o (0001)

(0001)1120

2.9 ± 0.7

C36 NbCo2

3

39o (0001)

(0001)1100

3.0 ± 0.5

C15 NbCo2

0.8

123 or 135

{111}110

4.6 ± 0.4

C15 NbCo2

0.8

100

{111}112

4.6 ± 0.5

C15 NbCo2

3

123 or 135

{111}110

3.2 ± 0.7

C15 NbCo2

3

100

{111}112

3.0 ± 0.8

C14 NbCo2

0.8

37o (0001)

(0001)1100

4.2 ± 0.5

C14 NbCo2

3

35o (0001)

(0001)1120

2.7 ± 0.1

C14 NbCo2

3

37o (0001)

(0001)1100

2.6 ± 0.5

Primary slip system CRSS [GPa]

* Only one 0.8 μm-sized micropillar of C36 NbCo 2 with 39o (0001) orientation was tested.

33

Table 5 The CRSS of the 3 µm-sized micropillars of the hexagonal C36 and C14 NbCo 2 Laves phases oriented for basal slip, prismatic slip {1100}1120 and pyramidal partial dislocation slip ( {1012}1011 for C36 and {1011}1012 for C14). Phase

Orientation

C36 NbCo2 36o (0001) or 39o (0001)

Primary slip system

CRSS [GPa]

(0001)1120 or (0001)1100

3.0 ± 0.6

C36 NbCo2

9o (0001)

{1100}1120

2.7 ± 0.1

C36 NbCo2

84o (0001)

{1012}1011

2.5 ± 0.2

(0001)1120 or (0001)1100

2.6 ± 0.4

C14 NbCo2 35o (0001) or 37o (0001) C14 NbCo2

6o (0001)

{1100}1120

2.6 ± 0.3

C14 NbCo2

69o (0001)

{1011}1012

2.5 ± 0.1

34

Figures and figure captions

Fig. 1 Microstructures and the corresponding concentration profiles of the (a) Co/Co-30Nb and (b) Nb/Co-30Nb diffusion couples. The concentration profiles were measured perpendicular to the interfaces of the diffusion layers by EPMA scans. Three EPMA scans were measured for each diffusion couple. The structure types of the NbCo2 Laves phase across the diffusion zones were identified by EBSD.

35

Fig. 2 Representative post mortem SEM images (side views after 54o tilt) of the single-slip (

123 or 135 ) orientated micropillars of C15 NbCo 2 having (a) 0.8 µm, (b) 3 µm and (c) 5 µm in top diameter and the 100 orientated micropillars of C15 NbCo 2 having (d) 0.8 µm, (e) 3 µm in top diameter. (f) Representative engineering stress-strain curves of the single-slip and 100 orientated C15 NbCo 2 micropillars. The corresponding top views of the micropillars and schematics of crystal orientation are inserted. The activated slip systems are indicated by arrows.

36

Fig. 3 Representative post mortem SEM images (side views after 54o tilt) of (a) 0.8 µm- and (b) 3 µm-sized 36o (0001) micropillars of C36 NbCo 2 orientated for (0001)1120 slip; (c) 0.8 µmand (d) 3 µm-sized 39o (0001) micropillars of C36 NbCo 2 oriented for (0001)1100 slip; 3 µmsized (e) 84o (0001) and (f) 9o (0001) micropillars of C36 NbCo2 oriented for non-basal slip. (g) Representative engineering stress-strain curves of C36 NbCo2 micropillars oriented for basal and non-basal slip. The corresponding top views of the micropillars and schematics of crystal orientation are inserted. The activated slip systems are indicated by arrows. 37

Fig. 4 Representative post mortem SEM images (side views after 54o tilt) of (a) 0.8 µm- and (b) 3 µm-sized 37o (0001) micropillars of C14 NbCo2 orientated for (0001)1100 slip; (c) 3 µmsized 35o (0001) micropillars of C14 NbCo2 oriented for (0001)1120 slip; 3 µm-sized (d) 69o (0001) and (e) 6o (0001) micropillars of C14 NbCo2 oriented for non-basal slip. (f) Representative engineering stress-strain curves of C14 NbCo2 micropillars oriented for basal and non-basal slip. The corresponding top views of the micropillars and schematics of crystal orientation are inserted. The activated slip systems are indicated by arrows. 38

Fig. 5 (a) TEM bright field micrograph of an [010] oriented micropillar of C15 NbCo 2 showing

(111)[121] slip (yellow lines) after deformation; (b)-(e) selected area diffraction patterns (SADP) taken along [101] zone axis of the corresponding regions. The yellow lines indicate the trace of the (111) plane and the red circles show the regions used for SADP. A post mortem SEM image of the [010] oriented micropillar is inserted.

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Fig. 6 (a) HAADF-STEM image of an [010] oriented micropillar of C15 NbCo 2 after deformation viewed along the [101] direction showing a stacking fault. (b) Schematic of the stacking sequence of the C15 NbCo2 Laves phase and (c) the stacking sequence of the C15 NbCo2 Laves phase with a stacking fault lying on the Z’ layer. The NbCo2 Laves phase is formed by stacking of a quadruple structural unit X (or Y, Z) which is composed of a single layer of Co atoms A (or B, C) and a sandwiched Nb-Co-Nb triple layer αcβ (or βaγ, γbα). The X’, Y’ and Z’ layers are twinned related to the X, Y and Z quadruple layers, respectively [3].

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Fig. 7 The size scaling of the CRSS of the C14, C15 and C36 NbCo2 micropillars. The 95 % confidence regions for the three Laves phases are marked in the figure.

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Fig. 8 Stereographic triangle exhibiting the orientations of the (a) single-slip ( 123 or 135 ) and (c) 100 orientated micropillars of the C15 NbCo2 Laves phase. The sketches in (b) and (d) show the relative Schmid factors between the leading and trailing partial as well as the corresponding deformation mechanisms of the C15 NbCo2 micropillars with single-slip and

100 orientation under uniaxial compression. See text for details.

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Fig. 9 The theoretical shear stress of the circular NbCo2 micropillar τpillar ( pillar  c theoretical ) for dislocation nucleation as a function of stacking fault energy γ and shear modulus μ. A schematic showing the nucleation of an incomplete partial dislocation loop (c = 0.4) at the surface of a circular micropillar is inserted.

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Fig. 10 The experimental CRSS measured from the 0.8 µm- and 3 µm-sized NbCo2 micropillars and the theoretical shear stresses of the NbCo2 micropillars for dislocation nucleation as a function of composition. The error bars of the experimental CRSS represent the standard deviations. The error bars of the compositions obtained from the concentration profiles represent the minimum and maximum compositions measured at the corresponding positions. The error bars of the theoretical shear stresses are based on the standard deviations of the shear moduli obtained from nanoindentation tests [18].

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Graphical abstrract

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