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Perturbed angular correlation studies of uniaxial compressive stressed zinc, titanium, rutile, Ti2AlN, and Nb2AlC

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 295501 (9pp)

doi:10.1088/0953-8984/26/29/295501

Perturbed angular correlation studies of uniaxial compressive stressed zinc, titanium, rutile, Ti2AlN, and Nb2AlC C Brüsewitz1, U Vetter1, H Hofsäss1 and M W Barsoum2 1

  II. Physikalisches Institut, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany   Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 19104, USA

2

E-mail: [email protected] Received 28 March 2014, revised 14 May 2014 Accepted for publication 30 May 2014 Published 24 June 2014 Abstract

We use the perturbed angular correlation method with 111In-111Cd probe atoms to in situ study the changes in the electric field gradient at room temperature of polycrystalline Ti2AlN and Nb2AlC, titanium and zinc, and rutile samples, as a function of cyclic uniaxial compressive loads. The load dependence of the quadrupole coupling constant νQ was found to be large in titanium and zinc but small in Ti2AlN, Nb2AlC and rutile. Reversible and irreversible increases in the electric field gradient distribution widths were found under load and after releasing the load, respectively. Annihilation of dislocations, as well as elastic deformation, are considered to contribute to the reversible behavior. The irreversible response must be caused by a permanent increase in dislocation and point defect densities. The deformation induced broadening of the electric field gradient can be recovered by post-annealing of the deformed sample. Keywords: perturbed angular correlation, compression, deformation, MAX phases, gradient elastic constants (Some figures may appear in colour only in the online journal)

1. Introduction

nanoindenter, for example—it is possible to nucleate nonbasal plane dislocations [4]. Dislocations are mobile at room temperature and arrange themselves either in walls normal to, or pile-ups parallel to, the basal plane [5]. Deformation is based on slip and delamination along basal planes, in combination with shear and kink band formation. The observation of a fully reversible and spontaneous dislocation-based deformation is believed to be due to the formation of incipient kink bands (IKB) [6, 7]. In their simplest form, IKBs consist of coaxial dislocation loops that nucleate under applied stress and reversibly annihilate upon unloading, unless the loops break through obstacles and dissociate into mobile dislocation walls which eventually form kink bands. Direct evidence for IKBs, and particular evidence of homogeneous dislocation nucleation, is still lacking. Recently, Jones et al [8] suggested that conventional dislocation flow in grains with appropriately oriented basal planes, in combination with elastic deformation of grains with other orientations, suffices to cause reversible hysteresis.

Perturbed angular correlation spectroscopy (PAC) provides, like other hyperfine interaction methods, information about the local structure around probe atoms in solids [1]. High sensitivity to small changes in the local structure around a probe atom promises insights into the remarkable mechanical properties, especially the damage tolerance and stability, of MAX phases [2]. The latter are a class of layered, machinable carbides or nitrides with the general structural formula Mn + 1AXn (n = 1, 2, 3), where M is an early transition metal, A is a subset of groups 13–16, and X is carbon or nitrogen. The mechanical properties of MAX phases have been summarized recently [2] and are attributed to their layered, hexagonal structure, strong M-X bonds on the one hand, and relatively weak M–A bonds on the other. Because their c/a-ratios are greater than 3.5, dislocations are almost entirely confined to the basal plane, with a Burgers vector of 1 / 3 〈 1120 〉 [3]. However, under extreme stress—under a 0953-8984/14/295501+9$33.00

1

© 2014 IOP Publishing Ltd  Printed in the UK

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Phase purity was verified with x-ray diffraction (D8 Advance of Bruker AXS). In order to introduce the PAC probe atoms into the samples, about 1012 111In ions were implanted at an energy level of 400 keV [30], corresponding to a mean penetration depth of about 100 nm, in an area of about 10 mm2. 111In decays within a half-life of 2.81 d into 111Cd, which acts as the probe atom. Lattice-site population and reduction of implantation damage were achieved by annealing Ti2AlN and Nb2AlC for five hours at 1173 K [24], Zn for an hour at 573 K [26], and Ti for five minutes at 473 K in vacuum. Rutile samples were obtained by surface oxidation of an already implanted Ti sample for five hours at 1273 K in a quartz ampule. PAC spectra were recorded at room temperature by a conventional four-detector setup equipped with NaI(Tl)scintillators. The implanted base area of the samples was aligned parallel to the detector plane. The corresponding experimental perturbation functions R(t) were extracted with SpectraPAC [31] and analyzed with Winfit [32]. A uniaxial load was applied parallel to the one-millimeter edges of the sample, and thus perpendicular to the detector plane, with a pneumatic press (PFW 056 of TOXPressotechnik) equipped with punches made of hardened steel. By measuring forces and the initial base area of the sample, engineering stresses σ were estimated. Permanent deformations εp were determined ex-situ by measuring the increase of the base area after removing the load on the assumption of volume conservation. The signs of compressive strains are defined to be positive. True stresses were approximated by σt = σ (1 − εp). The obtained PAC spectra were recorded in a quasi-static state while keeping a certain load constant for about 24 h. Two different loading procedures were used. The samples were either loaded to a certain stress and afterwards unloaded, which was repeated for different stresses, or loaded with successively increasing, and afterwards decreasing, stresses.

As noted above, the PAC method is sensitive to the local electric field gradient (EFG), a traceless second-order tensor determined by the second spatial derivatives of the electrostatic potential, at the nucleus of a probe atom. The EFG is obtained by measuring the time-dependent angular correlation of two photons emitted by a γ-γ-cascade. Only non-cubic charge distributions around the probe atom can give rise to a non-zero EFG. The method is suitable to identify point defects, which are trapped at the probe atom, by their characteristic EFGs [9–11]. Randomly distributed defects induce a distribution broadening of the field gradient tensor components by their strain fields, as seen in cubic and hexagonal metals [12, 13]. In contrast to the above-mentioned PAC experiments, which have been performed after deformation, in-situ experiments have only been conducted in order to investigate the effect of elastic strains on the EFGs [14–19]. These experiments were done under uniaxial compression of single crystals in the elastic regime or under hydrostatic compression. No broadening effects were observed. In general, similar questions have been studied with nuclear magnetic resonance spectroscopy (NMR) in cubic systems. A broadened distribution of the EFG tensor components around zero was observed after deformation from which dislocation densities were estimated by assuming a linear relationship between the EFG tensor components and the elastic stresses or strains [20, 21]. Previous work on MAX phases, using PAC with an 111 In-111Cd probe, revealed the existence of A site specific, strong, and axially symmetric EFGs in In, Al, Ge, and As containing 211-MAX phases [22–24]. Based on those results, it is now possible to investigate the deformation of polycrystalline MAX phases by an experiment, in which PAC spectra, measured under uniaxial load and after removing the load, are compared. In this respect, two representatives of the MAX phases, Ti2AlN and Nb2AlC, are systematically compared with other polycrystalline hexagonal metals, namely Ti [25], and Zn [26], as well as the tetragonal oxide rutile (TiO2), a system that exhibits an asymmetric EFG at the Ti-site [27]. Out of this, the pressure and deformation dependence of axially and non-axially symmetric EFGs of the polycrystalline materials under uniaxial load are studied.

3.  Data evaluation In general, the EFG can be parametrized by two quantities, the largest component of the diagonalized EFG tensor Vzz and the asymmetry parameter η = (Vxx − Vyy)/Vzz, where (x, y, z) is the principal coordinate system of the EFG tensor. An analytical expression of the perturbation function R(t), describing the interaction between the EFG and the quadrupole moment Q of a nuclear state with spin I, is given for polycrystalline cases as a function of η and Vzz in [33, 34]. The strength of the interaction is determined by the quadrupole frequency

2.  Experimental details The processing details of Ti2AlN and Nb2AlC can be found elsewhere [28, 29]. Briefly, stoichiometric ratios of Ti, and AlN or Nb, Al, and C powders were mixed and sintered in a hot press (Ti2AlN: 4 h at 35 MPa and 1773 K; Nb2AlC: 10 h at 35 MPa and 1873 K), resulting in predominantly single-phase samples, with grain sizes several tens of microns in diameter. Furthermore, bulk pieces of commercially available, 99.99% pure Ti and Zn (Alfa Aesar), with grain sizes of several tens and hundreds of microns respectively, were used. One millimeter-thick slices were cut out of bulk samples by spark erosion and divided into pieces with a 5  ×  6 mm2 base area. Parallel planes were achieved by grinding the surfaces with SiC sandpaper progressively finer down to 6 µm particle size.

eQVzz ωQ = (1) 4I (2I − 1) ℏ

or the quadrupole coupling constant νQ = 20/π·ωQ (for I = 5/2). Slight deviations from an ideal EFG give rise to an attenuation of R(t). The assumption of a distribution in only Vzz [35] leads to the widely used expression for the perturbation function 2

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⎛ ( g ωQδt )a ⎞ n ⎟⎟ , R ( t ) = A22 ∑ s2n cos( gnωQt ) exp ⎜⎜ − (2) a ⎝ ⎠ n=0

From the Czjzek-plots of the hexagonal systems Ti2AlN (see figure  1), Nb2AlC, Ti, and Zn (not shown) it was concluded that axial symmetry of the EFG indeed is preserved in all measurements. The maximum correlation is located at η  =  0; any deviation, visible as elongated areas, is assumed to be symmetric around zero (cf. interchange of Vxx and Vyy in the definition of η). In contrast, a fit to equation (2) yielded a nonzero η in the case of sufficient broad distributions, especially for Ti (η  =  0.13(3)). In rutile (see figure  2), the maximum correlation is located at η  =  0.19(1) in all measurements, contrary to an increase of η obtained by fitting the PAC spectra to equation (2). The artifacts mentioned above are direct evidence for the limited validity of equation  (2). This comment notwithstanding, the broadening parameter δ is a reliable measure of disorder as it can be considered to be an approximation of both contributions, the distribution widths of Vzz and η. Hence, in addition to νQ, δ will mainly be discussed in the following. The determined values of νQ, η and δ of the annealed samples are given in table 1. All fractions were identified as substitutional sites of the probe replacing Al in Ti2AlC and Nb2AlC [24], Ti in TiO2 [41] and Ti [25], as well as Zn in Zn [26]. The large value of δ in Ti is believed to be connected to impurities, most likely oxygen, and/or residual implantation defects. For that reason, the analysis for Ti is limited to νQ in the following. The application of uniaxial compressive stress led to different deformation morphologies for the various materials. Due to the chosen plate-like geometry, it was possible to load all samples to high stresses, even though some were loaded beyond their ultimate compressive strengths, without being pulverized or leaking radioactive isotopes. Therefore, all values of εp given hereafter should not be interpreted as plastic strains but as a crude measure of permanent deformation. In the case of rutile, strains are not given as the oxide layer did not deform visibly. Exemplary PAC spectra of samples under load are also shown in figures  1 and 2. Unlike changes of νQ and δ, significant changes of texture, which would be visible as a systematic deviation of the amplitudes s2n, were not observed. No additional interaction frequencies, which would indicate a phase separation or a trapped defect, were observed either. The broadening of Vzz and η is visible by comparing the areas of high intensity in the Czjzek-plots of the unloaded and the loaded samples. The dependence of νQ on uniaxial compressive stress is shown in figure 3. Within the measuring range, the dependence is large for Ti and Zn but small for rutile, Nb2AlC, and Ti2AlN. Values of νQ after removing the load did not significantly differ from the values obtained immediately after annealing (not shown). Linear regression of the data yielded the relative stress dependencies d ln νQ / dσt = (d νQ / dσt ) / νQ0 listed in table  1. In the case of uniaxial compression of Zn, d ln νQ/dσt was determined to be  − 86(6) × 10 − 3 GPa − 1, which is in the order of  − 56(4) × 10 − 3 GPa − 1 found in the literature for quasi-hydrostatic pressure [16]. The discrepancy is presumably caused by the different loading conditions, i.e. uniaxial loading of a (textured) polycrystalline system. No such data exists for the other materials.

3

of a single fraction of probe surroundings with the distribution width δ, the amplitudes s2n and coefficients gn depending solely on η [33, 34], and the anisotropy of the γ-γ-cascade A22. For a = 1, Lorentzian, and for a  =  2, Gaussian distributions are obtained. This approach neglects: (i) the distribution of η and, (ii) the true distribution function due to distinct kinds of defects. Systematic errors in Vzz and η, obtained by fitting experimental data to equation  (2), are especially large for wide distributions of the tensor components and an ideal η of zero or one [36]. An alternate approach in data evaluation is the cross-correlation of experimental data with the ideal powder perturbation function within the Czjzek-plot [37, 38]. In this representation, a two-dimensional distribution of the η-Vzz-map is visualized. By determining maximum intensities, a correct assignment of Vzz and η is possible, without any knowledge of the shape of the distribution function. For some special cases, such as uniform distributions in the Czjzek-map, analytical powder perturbation functions are given in [39]. In order to reconstruct more complicated distribution functions, simulations of the EFG in multiple defect arrangements are required, and the PAC spectra have to be evaluated numerically [40]. Despite the inaccuracies of the simplified attenuated expression of equation (2), fitting this model to the experimental data presented herein was possible. This attempt provided an easy and robust way to quantitatively determine a parameter δ for disorder in the absence of appropriate theoretical models. By using a fixed value of a = 2, i.e. a Gaussian distribution, the degrees of freedom were reduced, and thus the parameter δ could be compared between the samples. Furthermore, by choosing a = 2 the central limit theorem was taken into account, as the final EFG distribution is determined by the convolution of specific defect distributions, which may be approximated by a normal distribution in total. The amplitudes s2n were kept variable in order to allow for texture in the samples. In addition, Czjzek-plots were used to determine the fundamental structure of the EFG. 4.  Experimental results The PAC spectra of the annealed samples all exhibited a single fraction of a non-vanishing EFG. From the two examples shown, Ti2AlN in figure 1 and rutile in figure 2, it is clear that the experimental PAC spectra of all materials are fitted reasonably well by equation (2) for small attenuations. For large attenuations, a better agreement is possible by using a more complicated function, which can be illustrated by using specific δn instead of a general δ (see figure 2). This improvement is a direct consequence of the η distribution and the η dependence of gn. A Taylor series of this dependence predicts bigger slopes of g1 compared to g2 and g3, which corresponds to larger values of δ1 compared to δ2 and δ3, as suggested by the fitted values. Nevertheless, this improvement was for the most part not required in the course of this work, and therefore not used as it would have incorrectly introduced the η distribution. 3

J. Phys.: Condens. Matter 26 (2014) 295501

(a)

C Brüsewitz et al

T i2AlN

0.12

(b)

(c)

Cd on Al-sit e

6

0.08

4 2

0.04 loaded t o 1.02(3) GP a

0.12

A(ω)

-R(t )

111

8

0 8 6

0.08

4 2

0.04 0

100

200

t ime (ns)

0

300

0

300

600

ω (Mrad/s)

900

Figure 1. (a) Experimental and fitted PAC-spectra R(t) of Ti2AlN recorded after annealing (top panels) and under load (bottom panels), (b) their Fourier transforms and (c) the corresponding Czjzek-plots. Both spectra were fitted quite well with equation (2). The Czjzek-plot shows no deviation of axial symmetry under load. The distribution width of the EFG tensor components, i.e. Vzz and η, increases under load, visible as a damping in R(t) and a broadening in the ωQ-η-map.

(a) 0.08

T iO 2

(b)

8

111

Cd on T i-sit e

(c)

6

0.04

4 2

0.08

loaded t o 0.36(1) GP a

A(ω)

-R(t )

0.00

0 8 6

0.04

4 0.00 0

general δ specific δi

100

200

t ime (ns)

300

2 0 0

300

600

ω (Mrad/ s)

900

Figure 2. (a) Experimental and fitted PAC-spectra R(t) of TiO2 recorded after annealing (top panels) and under load (bottom panels), (b) their

Fourier transforms and (c) the corresponding Czjzek-plots. The as-annealed spectrum fitted quite well with equation (2). Under load, the fit of the spectrum could be improved by assuming a specific δn for each transition frequency. The Czjzek-plot shows no deviation of η = 0.19(1) under load. The distribution width of Vzz and η increases under load, seen as a damping in R(t) and a broadening in the ωQ-η-map.

The distribution widths δ are shown in figures 4 and 5 for different cycles of loading and unloading. Each cycle corresponds to a measurement at stress σ and a measurement after removing that load. Insets plot the corresponding permanent deformation εp after applying a certain engineering stress. In all samples, an increase in δ was observed under load. For the first cycles (see figure 4(a)), δ returned to its initial value upon unloading. When the yield point was exceeded, however, δ dropped to an intermediate level. From these results it is thus clear that irreversible changes of δ only occur with the onset of permanent deformation: i.e. there must exist a strain-dependent, irreversible part A(εp). On the other hand, a stress-dependent, reversible part B(σt) must also exist (see figure 4). As can be seen from figure  4(a), significant permanent deformation did not occur up to 0.6  GPa in Ti2AlN. In the range below, studies were carried out on a new sample separately. Stress was applied successively up to 0.4  GPa and

removed afterwards. The observed increase of δ is linear and nearly fully reversible (see figure 6). In order to quantify the reversible part B(σt), detailed knowledge of the distribution functions involved, and the consequent convolution, would have been required. This was not the case. Nevertheless, a rough approximation was deduced by a simple subtraction of δ from the corresponding δ under load. It became apparent that the reversible part is proportional to σt with a slope dδ/dσt depending on the material (see figure 7 and table 1). After loading and unloading to the maximum stress given by the cycle with the largest load, the irreversible increase A(εp) remained constant as long as the reapplied stress was below maximum. Independent of the sample's history, the reversible part B(σt) was visible under load at all times (not shown). It was possible to recover the irreversible part by using the same annealing method which had been used to reduce the implantation damage (see table 2).

4

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1.1(1) 0.9(1) 4.9(2) 0.7(1) 1.2(1)

 − 0.9(5)  − 0.2(6) 22(2)  − 86(6) 5(3)

4.3(4) 3.8(8) — 55(10) 60(5)

1 0

1-3

0

10

1

2

3

a

-1 0.0

0.2

0.4

0.6

0.8

1.0

1.2

2

5

) ) p

A( ε

6

t

7

Figure 3. Relative changes of the quadrupole coupling constant νQ

of the various materials, tested as a function of uniaxial compressive stress. Values of νQ after removing the load are not shown as they did not significantly differ from the values obtained immediately after annealing. Strong stress dependencies of νQ are only observed in Ti and Zn. The relative stress dependencies dln νQ/dσt are listed in table 1.

0.0

1

Nb 2AlC

5 4

0.5

1

0

uniaxial compressive stress σt (GPa)

0

2 1

loaded t o σ unloaded

3

2

εp (%)

4

2

3

cycle number

4

5

Figure 4. Distribution width δ of the EFG in (a) Ti2AlN and (b) Nb2AlC for different cycles of loading and unloading. Each cycle corresponds to measurement at a stress σ and after removing that load. The inset shows corresponding permanent deformation εp after applying a certain stress. δ rises reversibly with stress σt and irreversibly with deformation. In the case of Ti2AlN, no significant deformation was observed up to 0.6 GPa (cycle #4).

5. Discussion

In the case of symmetry-conserving strains [15], the volume dependence ∂lnνQ/∂lnV and the structure dependence ∂lnνQ/∂ln(c/a) of the EFG are usually used, rather than the general tensor Sijkl. In spite of the differences between uniaxial and isostatic compression, there are similarities between both geometries, which will be used in the following. Any uniaxial load not along the EFG's symmetry axis gives rise to a change of the local η, in addition to a change of Vzz, as seen in single crystal experiments [19]. Here, multiple orientations of the compression axis to the orientation of the EFG have to be taken into account due to the polycrystalline nature of the samples, resulting in a broadening caused by inhomogeneous stresses. The average changes in Vzz, i.e. dlnνQ/dσt, and η are in principal related to Cijkl. As pointed out above, Cijkl is affected by two material-specific tensors, namely Sijkl and the compliance sijkl, that both determine magnitude and sign of dlnνQ/dσt. The dependence on sijkl implies that stiffer materials exhibit smaller values of dlnνQ/dσt, simply because, for a given stress, they will experience smaller elastic strains. This is in agreement with the tendency of the experimental values shown in table  1 since the Young's moduli (as a

In the following, the influence of uniaxial compression and different defects on both axially and non-axially symmetric EFGs, will be discussed. Possible contributions of the IKB mechanism will be discussed later on. A useful approach towards understanding the MAX phases' and other materials' responses to uniaxial load and deformation is the gradient-elastic tensor [20]. In this formalism, the EFG tensor is connected, via gradient-elastic constants Cijkl or Sijkl (i, j, k, l ∈ x, y, z), to the stress tensor σkl or the strain tensor εkl, respectively, in analogy to the elastic constants of the stiffnes tensor cijkl or the compliance tensor sijkl: Vij − Vij0 = ∑ Cijklσkl = ∑ Sijklεkl . (3) kl

4

0.33(1) GP a

0

1.0

0.16(1) GP a

Ti Zn T iO 2

1

B( σ

3 σ (GP a)

Nb 2AlC

dist ribut ion widt h δ (%)

relat ive change of νQ (%)

(b)

T i2AlN

2

30

cycle number

As determined from Czjzek-plots.

3

20

εp (%)

1.25(4) GP a

0 0 0 0 0.19(1)

GPa )

loaded t o σ unloaded

0.96(3) GP a

258.1(3) 243.5(3) 27.3(2) 133.3(3) 105.6(2)

(10

2

0.0

5

1.10(3) GP a

(%)

 − 1

4

0.5

0.65(2) GP a

Ti2AlN Nb2AlC Ti Zn TiO2

 − 3

1.0

0.89(3) GP a

(MHz)

d ln νQ/dσt dδ/dσt

3

T i2AlN

7

6

0.59(2) GP a

δ

1.5

0.30(1) GP a

η

dist ribut ion widt h δ (%)

νQ

4

0.16(1) GP a

(a)

η, and Gaussian distribution width δ of the prevailing fraction in the PAC spectra of annealed Ti2AlN, Nb2AlC, Ti, Zn, and TiO2. The relative stress dependencies dlnνQ/dσt and dδ/dσt under uniaxial compressive stress were obtained from linear regression of the data shown in figure 3 and figure 7, respectively.

σ (GP a)

Table 1. Quadrupole coupling constant νQ, asymmetry parameter

a

kl

The EFG tensor, in the absence of stress, is denoted as Vij0. The tensor under load Vij is not given in its principal coordinate system but in the coordinate system (x, y, z) of Vij0. Sijkl is connected to Cijkl via the elastic constants cijkl. Exact knowledge of the constants Cijkl or Sijkl would have been required for a quantitative analysis. However, this is beyond the scope of this paper: symmetry arguments are used for discussion instead. 5

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2

0.0

1

3

2

0

4

10

6 5

5

20

30

1

2

3

4

cycle number

146(6) MP a

106(6) MP a

75(7) MP a

48(5) MP a

29(5) MP a

1

5

6

1

2

1.2

T i2AlN loaded

1.1 0.1

0.2

0.3

engineering stress σ (GP a)

0.4

3

the axial symmetry of the EFG in the Czjzek-plot is preserved and that the η distribution is broadened under load. In general, this does not apply to non-axially symmetric 0 0 is different from V yy —i.e., η is expected to be EFGs, as V xx stress-dependent. Such a change of η should be visible as a shift of the maximum intensity in the Czjzek plot, which was not observed under load. Therefore, the change of η under load in rutile is believed to be small compared to the accuracy of the Czjzek plot. Defects in the vicinity of probe atoms locally alter the EFGs with their strain fields, and thus result in increases in δ upon their introduction, regardless of their true distribution function. In general, point defects and dislocations have to be considered here. The influence of grain boundaries is believed to be small since the grain sizes D were on the order of several microns. Their contribution δGB is proportional to 1/D and only relevant in nanocrystalline material [47]. Delaminations and cracks are not considered for the same reason but should contribute to δ as well. By assuming a random distribution of point defects and dislocations, the contributions δPD and δD are proportional to their densities ρPD and ρ D , respectively. This is a direct consequence of the strain field εkl(r, Θ), which is proportional to the distance r between defect and probe atom, with 1/r3 for point defects and 1/r for dislocations [21]. The strength of this coupling is determined by Sijkl. Note that it is not possible to distinguish between point defects and dislocations within the given statistics and in the absence of theoretical models. For edge and screw dislocations [48], the average over all angles Θ of εkl(r, Θ) yields an average strain of zero in all components. Thus, the expected net change of Vij, due to dislocations, is zero. Point defects may give rise to a net increase or decrease of Vij in their surroundings, depending on the sign and magnitude of their strain field. Such a net change was not observed after deformation within the accuracy of the method, as pointed out in the experimental section. Despite this observation, the introduction of point defects cannot be excluded.

0.48(2) GP a

0

0.24(1) GP a

1

0.36(1) GP a

3 2

1.3

Figure 6. Distribution width δ of the EFG as a function of applied stress. Up to 0.4 GPa, no macroscopic deformation was observed. δ increases nearly fully reversibly with the applied load. This behavior can be caused by elastic effects and by reversible dislocation nucleation and annihilation. Afterwards, the very same sample was loaded to 1.2 GPa in order to force a permanent deformation of the sample (see table 2).

loaded t o σ unloaded

4

1.4

0.0

T iO 2

0.14(1) GP a

dist ribut ion widt h δ (%)

0.1

Zn loaded t o σ unloaded

6

εp (%)

0

(b)

0.2

dist ribut ion widt h δ (%)

3

σ (GP a)

dist ribut ion widt h δ (%)

(a)

4

5

cycle number Figure 5. Distribution width δ of the EFG in (a) Zn and (b) TiO2 for different cycles of loading and unloading. Each cycle corresponds to a measurement at a stress σ and after removing that load. The inset in (a) shows corresponding permanent deformation εp after applying a certain stress. No inset is given in (b) since deformations could not be measured. Both materials show a large reversible increase of δ under load.

polycrystalline average of the elastic constants) of Ti2AlN, Nb2AlC, and rutile are roughly three times larger than those of Ti and Zn [42–46]. Beyond that, the individual influence of Sijkl has to be considered. The volume dependence of the EFG ∂lnνQ/∂ ln V in hexagonal metals ranges from  − 3 to  − 6 [15], corresponding to a strong increase of νQ with isotropic volume decrease. As a consequence, the constants Sijkl are expected to be large in Ti and Zn, which presumably amplifies the effect of sijkl. The negative sign of dlnνQ/dσt for Zn is believed to be connected to the structure of sijkl. A distinctly large value of s33 [46] (Voigt notation) causes a strong decrease of the c/a-ratio under load, and so a decrease of νQ (a linear dependence of the EFG is assumed in hexagonal metals with the c/a-ratio [15]). This contribution outweighs the volume dependence, ending in a negative sign of dlnνQ/dσt. In Ti, the c/a ratio is only a weak function of load and so the volume dependence prevails. In the case of axially symmetric EFGs, isotropy in the 0 0 V xx - V yy plane, and the thus resulting equivalences in Cijkl, end in an equivalent deviation of Vxx and Vyy around a new average 〈Vxx〉   =  〈Vyy〉, after transforming into the principal axis system. This is in accordance with the observation that 6

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Table 2. Gaussian distribution width δ in Ti2AlN, Nb2AlC, and

reversible change of δ (%)

3.0 2.5

T i2AlN

2.0

Zn T iO 2

Zn (samples corresponding to the referred figures) after (i) initial annealing, (ii) deformation at maximum stress, and (iii) further ­annealing. The deformation-induced increase of δ is recovered to the initial valuea.

Nb 2AlC

1.5

Distribution width δ after Ti2AlN Nb2AlC Zn

1.0 0.5 0.0 0.0

b

Figure 6 Figure 4(b) Figure 5(a)

annealing

deformation annealing

1.1(1)% 0.9(1)% 0.7(1)%

1.9(1)% 1.2(1)% 1.6(1)%

0.9(1)% 1.0(1)% 0.7(1)%

a

 No such data were recorded for rutile since the rutile layer peeled off during annealing.

0.2

0.4

0.6

0.8

1.0

1.2

b

 Loaded to a maximum stress of 1.2 GPa after the cycle shown in figure 6.

uniaxial compressive st ress σt (GP a)

This is in accordance to the observed irreversible increase of δ with maximum applied load (see figures 4 and 5(a)). In case of figure  4(a), the maximum flow stress was exceeded at about σ  =  1.2 GPa. In order to compensate the applied load, the already failed sample was squeezed until the base area was as large as required for the maximum flow stress of about σt  =  1.0  GPa. This deformation, of course, is still introducing defects, visible as a continuing increase of δ. The maximum flow stress was not exceeded for the other samples, visible as a continuous increase of σt (see e.g. figure 3). As pointed out above, it is not clear whether dislocations or point defects are responsible for the irreversible part. In the case of the MAX phases it is quite surprising that recovering of δ is possible at the relatively low temperatures (see table  2) since dislocations are usually in the form of arrays and/or low angle grain boundaries, which are typically quite stable. If dislocations are involved, then they are most probably homogeneously distributed, trapped by point defects or other dislocations. The unpinning of such dislocations could be responsible for the changes observed. Neither can the production of point defects in the A-planes, and their annealing, be discarded as a potential mechanism. In the case of the reversible increase B(σt), it is not possible to distinguish between elastic strain, dislocation pile-up, and the IKB mechanism (i.e. between the models proposed in [6 and 8]) without a quantification of the individual components. All contributions could lead to the nearly fully reversible broadening observed in figure 6 and to the reversible broadening observed in any other PAC spectrum recorded under compression. The general mobility of dislocations at low stresses, a basic condition for reversible dislocation nucleation and annihilation, is well known for MAX phases [49]. Regarding the specific dlnνQ/dσt and dδ/dσt (see table 1), it is assumed that high values of dlnνQ/dσt are accompanied by broad distributions of η and Vzz under load, here combined in the quantity dδ/dσt. For Ti2AlN, Nb2AlC, and Zn, this expectation applies, whereas for rutile an unexpected large value of dδ/dσt is observed. The huge difference may be explained by a stronger weighting of the η distribution due to the linear dependence of the coefficients gn around η  =  0.19, in contrast to a quadratic dependence around η  =  0 in the case of Zn and the MAX phases.

Figure 7. Reversible change of δ or B(σt) as a function of

uniaxial compressive stress for Ti2AlN, Nb2AlC, Zn and TiO2, determined by the difference between measured δ under load and after removing the load. Even though a linear dependence is an oversimplification, the slopes dδ/dσt are given in table 1 for comparison.

Possible phase transformations upon ion implantation or contaminants in the samples, e.g. carbides in the MAX phases, alter macroscopic mechanical properties: such as, yield strengths or elastic constants. However, the lattice sites examined here are solely the phase specific, substitutional sites mentioned above. Probes which reside on unintended lattice sites or in other phases would either be visible as new EFGs or as an offset of the perturbation function. The former was not observed and the latter does not affect the distribution width of the site specific EFG. Therefore, it can be concluded that all information taken from the distribution width is unaffected by any contaminants and only arises from defects within the specific phase. Thus, the following picture emerges: The initial dislocation and point defect densities, as well as the grain size, determine δ after annealing. Deformation-induced dislocations, and thereby introduced point defects (and to some extend cracks and delaminations, where applicable), give rise to an irreversible increase A(εp) in δ. Uniaxial load results in a reversible increase B(σt) of δ, which could be caused by contributions of uniaxial elastic strain in the polycrystalline material and dislocations that only exist under load, i.e. those associated with incipient kink bands [6]. Point defects can safely be excluded from contributing to the reversible part. The assumption that any irreversible increase of δ is connected to permanent deformation is reasonable. In order to cause macroscopic permanent deformation, a stress equal to at least the yield strength of the material must be applied. Any measure at a certain stress σt above the yield strength is a measure at the instantaneous flow stress. The observation of an increase of the flow stress (see insets in figures 4(a) and 5(a)) implies that work hardening is occurring, and is hence an indirect evidence for an increase in the density of dislocations. By using the proportionality of the flow stress to ρ D from dislocation theory and the ρ D dependence of δ, a linear dependence of δ as a function of stress is expected [21]. 7

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[4] Tromas C, Villechaise P, Gauthier-Brunet V and Dubois S 2011 Slip line analysis around nanoindentation imprints in Ti3SnC2: a new insight into plasticity of MAX-phase materials Phil. Mag. 91 1265–75 [5] Barsoum M W, Farber L and El-Raghy T 1999 Dislocations, kink bands, and room-temperature plasticity of Ti3SiC2 Metall. Mater. Trans. A 30 1727–38 [6] Barsoum M W, Zhen T, Kalidindi S R, Radovic M and Murugaiah A 2003 Fully reversible, dislocation-based compressive deformation of Ti3SiC2 to 1 GPa Nature Mater. 2 107–11 [7] Barsoum M W and Basu S 2010 Kinking nonlinear elastic solids Encyclopedia of Materials: Science and Technology 2nd edn ed K H Jürgen Buschow et al (Oxford: Elsevier) pp 1–23 [8] Jones N G, Humphrey C, Connor L D, Wilhelmsson O, Hultman L, Stone H J, Giuliani F and Clegg W J 2014 On the relevance of kinking to reversible hysteresis in MAX phases Acta Mater. 69 149–61 [9] Collins G S, Stern G P and Hohenemser C 1981 Vacancy trapping in plastically deformed metals studied by hyperfine interactions Phys. Lett. A 84 289–93 [10] Wichert Th 1983 Hyperfine interactions: a tool for the study of lattice defects Hyperfine Interact. 15 335–55 [11] Deicher M, Grübel G, Reiner W and Wichert Th 1987 Trapping of lattice defects at 111 In atoms in cold-worked copper Mater. Sci. Forum 15–8 635–40 [12] Wodniecka B, Marszalek M and Wodniecki P 1989 TDPAC studies on the recovery of Ag plastically deformed at room temperature J. Phys.: Condens. Matter 1 7521 [13] Rasera R L, Reno R C, Schmidt G, Butz T, Vasquez A, Ernst H, Shenoy G K and Dunlap B D 1978 Strength, symmetry and distribution of electric quadrupole interactions at 181Ta impurities in hafnium-zirconium alloys J. Phys. F: Met. Phys. 8 1579 [14] Ernst H, Butz T and Vasquez A 1977 The electric field gradient at tantalum in a hafnium single crystal subjected to hydrostatic and uniaxial compression J. Phys. F: Met. Phys. 7 1329 [15] Butz T 1978 On the volume and structure dependence of electric field gradients in close-packed metals. I Phys. Scr. 17 87 [16] da Jornada J A H and Zawislak F C 1979 Effects of high pressure on the electric field gradient in sp metals Phys. Rev. B 20 2617–23 [17] Marx G and Vianden R 1996 Electric field gradients in Si induced by uniaxial stress Phys. Lett. A 210 364–9 [18] Tessema G 2006 Uniaxial compressive stress induced nuclear quadrupole interaction at the 111Cd nucleus in n-doped silicon Physica B 373 28–32 [19] Przewodnik R, Kessler P and Vianden R 2013 Hyperfine fields in ZnO studied under uni- and biaxial pressure Hyperfine Interact. 221 111–6 [20] Shulman R G, Wyluda B J and Anderson P W 1957 Nuclear magnetic resonance in semiconductors. II. Quadrupole broadening of nuclear magnetic resonance lines by elastic axial deformation Phys. Rev. 107 953–8 [21] Kanert O and Mehring M 1971 Static Quadrupole Effects in Disordered Cubic Solids (NMR vol 3) ed P Diehl et al (Berlin: Springer) pp 1–81 [22] Jürgens D, Uhrmacher M, Hofsäss H, Röder J, Wodniecki P, Kulinska A and Barsoum M W 2007 First PAC experiments in MAX-phases Hyperfine Interact. 178 23–30 [23] Jürgens D, Uhrmacher M, Hofsäss H, Mestnik-Filho J and Barsoum M W 2010 Perturbed angular correlation studies of the MAX phases Ti2AlN and Cr2GeC using ion implanted 111In as probe nuclei Nucl. Instrum. Methods Phys. Res. B 268 2185–8

On the other hand, large values of dδ/dσt are not necessarily accompanied by high values of dlnνQ/dσt. It is conceivable that the new average 〈Vzz〉 is a weak function of σt, despite large values of the Cijkl components. Furthermore, sources of broadening have to be taken into account under load, which do not have to end in a net change of Vzz; namely, dislocations such as those supposed to exist in the IKBs. However, it is not possible to discuss any (non-linear) contribution of the IKB mechanism within the given statistics and without further knowledge of the convolution of all contributions to δ. A quantitative analysis, in order to distinguish between the involved contributions but also to determine defect densities, would require knowledge of Cijkl or Sijkl and the strain fields of individual defects, always with respect to the anisotropic symmetry. The gradient elastic constants could be determined either from compression and torsion of single crystalline samples, i.e. by measuring Cijkl, or from density functional theory calculations by studying Vij as a function of εkl, i.e. calculating Sijkl by varying the lattice parameters a, c, α and γ, in the case of hexagonal symmetry. Furthermore, the contribution of dislocations could be isolated by diffraction methods (see e.g. [50]). 6. Conclusion In summary, we have studied, for the first time, the effects of uniaxial compressive stress on the EFG in plastically anisotropic materials: namely, zinc, titanium, rutile, Ti2AlN, and Nb2AlC. We have shown that the PAC method is suitable for systematically investigating stress- and defect-induced changes in the electric field gradient. The change of the quadrupole coupling constant, under uniaxial load, was found to be larger in the hexagonal metals than in the MAX phases or rutile. We observed both reversible and irreversible broadening of the EFG components. Acknowledgments We would like to thank M Uhrmacher, D Jürgens, M Nagl, D Purschke, and C A Volkert from the University of Göttingen for help with the PAC measurements and fruitful discussions and D J Tallman, M Shamma, E N Caspi, and Z Nikolov from Drexel University for help with the sample preparation. This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) under contract no. HO 1125/19-2. M W Barsoum acknowledges the support of the Army Research Office (W911NF-11-1-0525) and the National Science Foundation (DMR-1310245). References [1] Frauenfelder H and Steffen R M 1965 Alpha-, Beta- and Gamma-Ray Spectroscopy ed K Siegbahn (Amsterdam: North-Holland) pp 997–1198 [2] Barsoum M W 2013 MAX Phases: Properties of Machinable Ternary Carbides and Nitrides (New York: Wiley) [3] Farber L, Barsoum M W, Zavaliangos A, El-Raghy T and Levin I 1998 Dislocations and stacking faults in Ti3SiC2 J. Am. Ceram. Soc. 81 1677–81 8

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