Scholarly article on topic 'Dominating deformation mechanisms in ultrafine-grained chromium across length scales and temperatures'

Dominating deformation mechanisms in ultrafine-grained chromium across length scales and temperatures Academic research paper on "Materials engineering"

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Acta Materialia
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Keywords
{"Strain-rate sensitivity" / "Ultrafine-grained materials" / "Thermally activated processes" / "Elevated temperature testing" / In-situ / "Scale-bridging experiments"}

Abstract of research paper on Materials engineering, author of scientific article — R. Fritz, D. Wimler, A. Leitner, V. Maier-Kiener, D. Kiener

Abstract The microstructure influence on the thermally activated deformation behaviour of chromium is investigated for a more fundamental understanding of the deformation mechanisms contributing to plasticity in bcc metals. Therefore, scale-bridging experiments at variable temperatures and varying strain-rates are performed, encompassing macroscopic compression tests in direct correlation to local in-situ SEM micro-compression experiments on taper-free pillars and advanced nanoindentation testing. For the first time, it is demonstrated that, independent of stress state, sample volume and surface fraction, a distinct temperature-dependent transition of the dominating deformation mechanism occurs. While at low temperatures the lattice resistance dominates, exceeding a critical temperature the dislocation interaction with grain boundaries becomes the rate limiting step. Finally, based on the vastly different fractions of grain boundaries in the tested sample volumes, a comprehensive model on the deformation of bcc metals, in particular at small scales or for confined volumes is derived.

Academic research paper on topic "Dominating deformation mechanisms in ultrafine-grained chromium across length scales and temperatures"

Accepted Manuscript

Dominating Deformation Mechanisms in Ultrafine-grained Chromium across Length Scales and Temperatures

R. Fritz, D. Wimler, A. Leitner, V. Maier-Kiener, D. Kiener

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Reference: To appear in:

S1359-6454(17)30698-5 10.1016/j.actamat.2017.08.043 AM 14004

Acta Materialia

Received Date: Revised Date: Accepted Date:

23 May 2017 26 July 2017 21 August 2017

Please cite this article as: R. Fritz, D. Wimler, A. Leitner, V. Maier-Kiener, D. Kiener, Dominating Deformation Mechanisms in Ultrafine-grained Chromium across Length Scales and Temperatures, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.08.043

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Dominating Deformation Mechanisms in Ultrafine-grained Chromium across Length Scales and Temperatures

R. Fritz1, D. Wimler1, A. Leitner1, V. Maier-Kiener2, D. Kiener1,*

1 Department Materials Physics, Montanuniversität Leoben, Leoben, Austria

2 Department of Physical Metallurgy and Materials Testing, Montanuniversität Leoben, Leoben, Austria

*Corresponding Author

Abstract

The microstructure influence on the thermally activated deformation behaviour of chromium is investigated for a more fundamental understanding on the deformation mechanisms contributing to plasticity in bcc metals. Therefore, scale-bridging experiments at variable temperatures and for varying strain-rates are performed, encompassing macroscopic compression tests in direct correlation to local in-situ SEM micro-compression experiments on taper-free pillars and advanced nanoindentation testing. For the first time, it is demonstrated that, independent of stress state, sample volume and surface fraction, a distinct temperature-dependent transition of the dominating deformation mechanism occurs. While at low temperatures the lattice resistance dominates, exceeding a critical temperature the dislocation interaction with grain boundaries becomes the rate limiting step. Finally, based on the vastly different fractions of grain boundaries in the tested sample volumes, a more comprehensive model on the deformation of bcc metals, in particular at small scales or for confined volumes is derived.

Keywords

Strain-rate sensitivity, ultrafine-grained materials, thermally activated processes, elevated temperature testing, in-situ, scale-bridging experiments

1. Introduction

Over the last decades, investigations on the elemental deformation mechanisms in different metals, in particular body-centred cubic (bcc) ones, were extended from originally coarse-grained (cg) states [1,2] to single-crystalline (sxx) [3,4], ultrafine-grained (ufg) [5-7] and even nano-crystalline (nc) [8,9] microstructures. Since macroscopic tests average deformation characteristics over several length scales, testing of limited sample volumes [10] offers the premise to assess specific intrinsic material behaviour. In these small dimensions, individual plasticity mechanisms, such as dislocation motion, dislocation interactions, dislocation pile-ups [9], or even diffusion-mediated processes such as grain boundary (GB) sliding [11,12] can potentially be isolated, thereby allowing identification and analysis of specific deformation modes.

A common approach to gain insight into the thermally activated deformation behaviour of bcc metals is to determine the strain-rate sensitivity (m) and the correspondent activation volume (v) [13]. Therefore, constant strain-rate tests or/and strain-rate jump (SRJ) tests are conducted and the rate-dependent stress responses are used to determine m. Since in this work the aim is to bridge from macroscopic experiments to small volumes, it is important to note that recent work on nc Ni, ufg Al and ufg Nb [14-16] revealed direct comparability of constant strain-rate tests and SRJ tests performed by nanoindentation, small-scale tension and compression experiments with macroscopic data.

Conventional cg or sxx face-centred cubic (fcc) metals exhibit m-values in the order of 10-3 [5,17-20]. If internal length scales for dislocation interaction become smaller, e.g. by decreasing the grain size, m increases by about one order of magnitude [9,17]. The corresponding v (indicated in multiples of the cubed Burger's vector b) decreases from values well above 100 b3 (cg microstructure) to a couple of 10 b3 in ufg fcc metals [9,18]. This indicates a transition from forest dislocation interactions to dislocation-GB interactions. Further

decreasing the grain size to the nc regime leads to low v of ~1 b3. These values are classically attributed to diffusion-driven processes [9].

To investigate the GB contribution in confined sample volumes, Zhang et al. [19,20] performed micro-compression tests on sxx and ufg Cu pillars and reported that m is strongly dependent on the sample diameter to grain diameter (D/d) ratio. While sxx and macroscopic polycrystalline Cu samples show low m-values of ~0.002, they reported ~0.10 - 0.15 for D/d ranging between 3 and 10. Thus, the high number of interfaces in the deformed volume strongly influences the deformation behaviour [5,9,21] as well as the yield stress [22-25] of pillars in the fcc case.

While the situation in fcc metals is quite well understood, the situation is less clear in bcc structures [26,27]. This prevails especially for cases where microstructure, microstructural constraints, loading conditions and variable sample sizes are taken into account. Moreover, the flow stress in bcc metals consists of two parts, namely the temperature independent athermal component (aa), which arises from long-range stresses caused by obstacles to dislocation motion, such as impurities or GBs, and the temperature and strain-rate dependent thermal component of the flow stress (ath), which stems from the resistance of the lattice itself, called the Peierls potential [1,28]. In the latter case, the movement of screw dislocations via the kink-pair mechanism becomes the dominating thermally activated contribution during low temperature deformation [29].

The temperature dependency varies with respect to a critical material specific temperature (Tc), upon which thermal activation eases the movement of screw dislocations with increasing temperature. Eventually, the lattice friction diminishes once Tc is reached at ~0.2 • Tm [29], with Tm being the melting temperature of the respective metal. Above this temperature (for Cr ~160°C [29]), the rate-dependent characteristics in bcc and fcc metals are comparable since the Peierls potential contribution vanishes. Therefore, screw dislocations

exhibit a similar mobility as edge dislocations [28], which consequently leads to low m and corresponding high v typical for fcc metals, as only long range stresses prevail.

In literature, investigations on bcc metals were conducted by Wei et al. [30-34] performing macroscopic compression tests and Zhou et al. [35] and Wu et al. [8] performing nanoindentation tests, addressing the rate-dependent deformation behaviour on V, W, Mo, Ta and Cr. Increasing m with increasing grain size was reported, opposite to reports on fcc metals [5,9,21].

More recently, Maier et al. [6,7] investigated the deformation mechanisms in sxx and ufg bcc metals by means of advanced nanoindentation techniques at room temperature (RT) for Cr and W [7], and at elevated temperatures for Cr [6]. For Cr, m at RT of ~0.07 in sxx samples was attributed to a strong contribution of the Peierls potential and a low mobility of screw dislocations [28] which govern the deformation process at low homologous temperatures underneath Tc via the kink pair mechanism [1,28,29]. Comparably lower m at RT of ~0.02 in ufg samples was referred to a prevailing contribution of the Peierls potential in combination with an increased athermal contribution due to GB strengthening. Overcoming Tc in the ufg state, a further increase of m was measured and related to a diminishing contribution of oth accompanied by an emerging dominant thermally activated dislocation-GB interaction. oa remains mostly constant with increasing temperature due to a thermally stable microstructure [6].

In this work, focus is placed towards a more comprehensive scale-bridging understanding of the deformation characteristics of bcc Cr by examining contributing factors such as microstructure, sample size, temperature and stress state. The corresponding effects on the deformation behaviour over four orders of magnitude concerning the sample size, taking into account rate effects at ambient as well as non-ambient conditions, are analysed in this study. Therefore, uniaxial macroscopic compression tests, in-situ SEM micro-compression experiments, as well as multiaxial advanced nanoindentation experiments using different tip

geometries were performed at variable strain-rates between RT and 400°C to determine mvalues and activation volumes. The stress-strain response and occurring deformation mechanisms were compared with sxx Cr to assess the impact of GB contributions. Moreover, the rate-dependent properties and microstructural evolution will be related to the loaded material volume with respect to testing temperature (Ttest) and fraction of GBs present within the specimen to assess size-dependent mechanism transitions.

2. Material Processing

The as-received polycrystalline ultra-high-purity Cr sheets (Cr-265, Plansee SE, Reutte, Austria) were cut by Electrical Discharge Machining (EDM, Brother HS-3100) to a cylinder with a diameter of ~30 mm and a height of ~7 mm. Subsequently, this cylinder was deformed via High Pressure Torsion (HPT) [36,37] using a rotational speed of 0.5 rpm and a pressure of 4.2 GPa at 200°C to an equivalent strain of ~360 (50 rotations) to reach an ufg microstructure. Subsequently, the Vickers hardness (Buehler MicroMet 5104, load of 500 gf) of ufg Cr was measured over the whole disk radius (r) and disk thickness on polished samples and is shown in Figure 1a on top of the half HPT disk.

2.1 Compression Testing

The EDM-cut macroscopic compression samples indicated in Figure 1a were subsequently ground and polished with SiC paper to achieve a smooth sample surface with final sample dimensions of 2 • 2 • 3 mm3. They were machined in axial disc direction, where elongated grains from the HPT process are oriented perpendicular to the compression axis. The samples were held between two WC-Co plates and sample-plate interface friction during deformation was neglected. Testing was performed in air using a universal tensile testing unit (Zwick GmbH & Co KG, Ulm, Germany) modified with a load-reverse tool to a compression

device. Loads were measured with a calibrated 10 kN load cell and strains were calculated from recorded time and the corresponding crosshead velocity.

All samples were compressed to ~20% plastic strain. Testing temperatures (Ttest) between RT and 400°C (corresponding to 0.69 • Tc up to 1.57 • Tc, where Tc = 160°C [29]) were applied for all testing techniques. Within this temperature range, creep mechanisms that may occur above TtestITm > 0.4 are minimized, since the range from RT to 400°C corresponds to 0.14 - 0.31 • Tm. Constant strain-rates between 10-2 s-1 and 10-4 s-1 were applied to the displacement-controlled tests to ensure appropriate force-displacement data and reduce measurement times to minimize thermal drift influences. The flow stress at ~8% plastic strain was evaluated to allow comparison with nanoindentation tests performed with a Berkovich indenter and to include the strain hardening behaviour within the first few percent of deformation to be comparable with previous bcc studies [38-44].

For pillar-compression experiments, a 3 • 2 • 1 mm3 sized sheet was cut in axial direction from the HPT disk at r~14 mm by EDM (Figure 1a), and subsequently thinned and polished to a lamella. Rectangular non-tapered pillars in the size range of 0.2 um to 6 um and with an aspect ratio of 2.5 - 3 were milled using a dual-beam SEM-FIB (Zeiss LEO 1540 XP, Oberkochen, Germany) [45]. Sxx pillars (crystal obtained from Mateck GmbH, Jülich, Germany) were prepared the same as the ufg pillars, and the (100) crystal direction was chosen to be the compression direction. The micro-compression tests were carried out in-situ in an SEM (Zeiss LEO 982, Oberkochen, Germany). Pillars smaller than 1 um were tested with a Hysitron PI85 Picoindenter® using a feedback loop of 200 Hz, while for pillars larger than 1 um, an UNAT-SEM indenter (Zwick GmbH & Co. KG, Ulm, Germany) with a feedback loop of 64 Hz was utilized, as this device offers the required higher loads. For RT, the indenters were equipped with conductive diamond flat punches (Synton-MDP AG, Nidau, Switzerland) with diameters of ~6 um and ~8 um, respectively. Stress-strain curves were calculated from the recorded force-displacement data after taking into account the machine stiffness. The top pillar

area was used for stress calculations, and the height of the untapered samples was applied to calculate strains [46].

2.2 High temperature pillar-compression

For micro-pillar compression tests at elevated temperature, a custom made, resistive

heating device equipped with a ~12 |im sapphire flat punch (Synton-MDP AG, Nidau, Switzerland) was installed on the UNAT-SEM indenter. To achieve isothermal contact temperatures, the sample holder was also independently heated on the moveable stage of the SEM. Thermocouples were brazed nearby the indenter and the sample holders. Software control was implemented in LabView® (National Instruments Corp., Austin, Texas, USA) and temperature control was achieved with a PID feedback loop. With this setup, accurate measurements at temperatures up to 300°C are achievable without active cooling after a stabilisation time of ~30 min. For temperature calibration of the equipped indenter, a temperature matching procedure as suggested in [47] has been applied to tune the contact temperature between the indenter and the sample in order to minimize thermal drift. Further details about the setup are summarized in [48].

2.3 Nanoindentation

HPT-deformed macroscopic ufg samples as well as an sxx sample were mechanically and electrolytically polished to remove remaining deformation layers before nanoindentation testing. In order to measure thermally activated processes at various temperatures, nanoindentation strain-rate jump (SRJ) tests [14] were performed using a Nanoindenter G200 (Keysight Technologies, USA) equipped with a continuous stiffness measurement (CSM) unit to continuously record contact stiffness and to avoid local thermal drift [49]. Machine stiffness and tip shape calibrations were performed at RT according to the Oliver-Pharr method [50]. Strain-rate jumps for displacement segments of 500 nm each and strain-rate levels of 5 • 10-2 s-1, 10-2 s-1, 10-3 s-1 and 5 • 10-3 s-1 were performed. Furthermore, strain-rate controlled tests with

a constant strain-rate of 5 • 10-2 s-1 were conducted to compare to results from SRJ testing as well as uniaxial testing techniques. For all tests, the CSM frequency was set to 45 Hz and a harmonic displacement amplitude of 2 nm was superimposed.

For RT testing, a three-sided diamond Berkovich pyramid (imposing ~8% plastic strain, obtained from MicroStar Technologies, Huntsville, USA) as well as a diamond Cube Corner indenter (Synton-MDP AG, Nidau, Switzerland), which imposes ~20% plastic strain [51], were utilised. For high temperature measurements at 100°C, 150°C, 200°C, 250°C and 300°C a Berkovich pyramid tip made of sapphire (Synton-MDP AG, Nidau, Switzerland) was used. Independent heating of sample and indenter was achieved by a laser heating stage (SurfaceTec, Hückelhoven, Germany) and an active water-cooling system. Moreover, the maximum indentation depth for all indents was 2500 nm and tests under non-ambient conditions were performed in an inert gas environment (95% N2 and 5% H2) to exclude sampling issues and oxidation effects.

3. Results

3.1 Microstructure and hardness

The initial grain size of the as-received polycrystalline Cr was ~200 |im (Figure 1b, left). The mean grain size in the ufg lamella used for pillar preparation was ~160 ± 51 nm (Figure 1b, middle) and in the macroscopic compression samples, ranging from r~12 mm to r~14 mm ~300 ± 86 nm (Figure 1b, right). In both cases, grains are slightly elongated but no pronounced texture was observed. The average aspect ratio of elongated grains was 2.7/1 ± 0.09 in axial direction, and no pronounced substructure formation was observed.

Hardness values are presented in Table 1 and as colour code in Figure 1. Due to the radial strain gradient in the HPT device and in accordance with the microstructural variations, a slight hardness change was observed. Hardness values of 4.40 ± 0.10 GPa at r ~14 mm and 4.20 ± 0.10 GPa at r~12 mm were measured. A hardness deviation along the disk thickness

(axial direction) of ± 0.10 GPa was observed at r~14 mm and ± 0.03 GPa at the centre of the disk. The change along the disk radius (radial direction) between the disk centre and ~14 mm is ± 1.00 GPa.

3.2 Stress-strain response

Exemplary engineering stress-strain curves of pillars and macroscopic compression samples are shown in Figure 2. The stress-strain responses with varying strain-rates for ~2 |im sxx and ufg pillars tested at RT are presented in Figure 2a. The flow stresses of sxx pillars are around 500 MPa, while for ufg samples an increase to ~2080 MPa at 8% plastic strain is evident. At the same time, the change in flow stress with varying strain-rate (high: ~2 • 10-2 s-1, medium: ~8 • 10-3 s-1, low: ~3 • 10-3 s-1) is significantly less pronounced in the ufg states compared to sxx pillars. Related nanoindentation hardness data obtained with a Berkovich indenter are included as horizontal lines. Comparison of stress-strain data between RT and 230°C (Figure 2b) reveals decreased flow stresses for sxx and ufg pillars, and a diminishing work hardening for the pillars at elevated temperature. Figure 2c shows the stress-strain response of macroscopic ufg samples tested at a constant strain-rate of 3 • 10-3 s-1 at different temperatures. The flow stress decreases with increasing temperature, and the strain hardening behaviour is less pronounced at elevated temperatures, in accordance with the micro-pillar data. Figure 3a represents exemplary constant strain rate nanoindentation load-displacement curves compared to SRJ tests for sxx (red) and ufg Cr (blue), respectively, showing good match between the two techniques for the same strain rate. Figure 3b depicts nanoindentation hardness and Young's Modulus values plotted vs. indentation depth. The mean Young's Moduli extracted for sxx and ufg Cr (303 GPa) are close to the literature value of 294 GPa [52]. Finally, Figure 3c represents exemplary hardness divided by a constraint factor (C*) of 2.8 vs. indentation depth plots for Cube Corner and Berkovich indentations in sxx and ufg Cr at RT, 200°C and 300°C. The strength in the ufg samples is generally higher due to grain refinement,

and a decreased hardness is observed with increasing Ttest. Differences between Cube Corner (20% representative strain) and Berkovich data (8% representative strain) at RT stem from the varying representative strain imposed to the material, which is conform to the uniaxial data. Moreover, the sxx indents exhibit a noticeable indentation size effect (ISE), which was accounted by analysing indentation data according to the model of Nix and Gao [53] to extract the bulk hardness H.

Post compression SEM images of macroscopic samples (Figure 4a) and micro-pillars (Figure 4b,c) reveal the appearance of the deformed samples corresponding to the stress-strain curves in Figure 2. Figure 4a shows macroscopic samples deformed at RT and a strain-rate of 2 • 10"2 s-1, RT and 3 • 10-3 s-1, and 400°C at a strain-rate of 2 • 10-3 s-1, respectively. Figure 4b represents ~3 |im and ~4 |im ufg pillar deformed at low strain-rates at RT and 230°C, respectively. A bulk-like deformation behaviour is observed at all tested temperatures and strain-rates. Due to small sample dimensions and a comparatively large grain size, near-surface grains appear to be pushed out of the surface. This behaviour is observed for all different temperatures, strain-rates and pillar sizes, respectively. Figure 4c shows sxx pillars deformed at RT (left) and 230°C (middle). The former shows deformation by crystallographic slip on ill-defined slip planes as expected for the (100) orientation, whereby the latter exhibit a more localized slip behaviour with sharper slip bands. The insets in Figure 4b and c provide details of the sample surfaces.

Figure 5 depicts residual indents of Berkovich and Cube Corner imprints compared with images obtained from in-situ pillar-compression at comparable uniaxial strain. Figure 5a presents indents in ufg Cr at RT and 300°C, while Figure 5b gives a comparison of indents in sxx Cr at the corresponding temperatures. Figures 5c and d show in-situ SEM images of ufg Cr pillars during compression at ~8% plastic strain and post compression at ~20% plastic strain tested at RT and 230°C, respectively. Slight pile-up and emergence of grains from the surface

is evident for the Cube Corner indents and coincides with the surface of the 20% deformed

pillars.

4. Discussion

As long as sample size effects in micro-pillars [10,54,55] or an ISE during nanoindentation [53] are properly accounted for, a comparison of different testing techniques should generally lead to comparable results, as shown for different non-textured ufg fcc materials [14,15,51,56]. For the ufg micro-compression experiments, this is taken into account by the smaller internal grain size length-scale dominating over the sample size effect [10,54,55]. The dominant dislocation character in sxx and ufg samples was considered to be of screw type, as suggested by Cheng et al. [57] for length scales larger ~300 nm. Regarding the nanoindentation data, this is considered by extrapolating to bulk hardness values from ISE affected nanoindentation data [53] as seen in Figure 3c. Moreover, pile-ups in the present case insignificantly influence hardness values, as indicated by the constant profiles of Young's Modulus as shown in Figure 3b. The choice of a reasonable constraint factor (C*) is important to link nanoindentation hardness to a corresponding uniaxial stress. Several values for C* are discussed in literature ranging from 2.5 to 3.0 for metals [14,51,58-64] to account for the multiaxial highly hydrostatic stress state during indentation. In the present work, 2.8 was chosen as proposed by Tabor [51].

4.1 Stress-strain characteristics

First, the stress-strain data (Figure 2 and 3) for different testing techniques will be discussed addressing RT behaviour (I), characteristics above Tc (II), followed by the temperature-dependent strain hardening behaviour (III), and the corresponding flow characteristics (IV).

(I) Stress values at 8% plastic strain in the ufg Cr pillars (~2080 MPa) and nanoindentation data (~2050 MPa) at RT are in good accordance, implying that global flow stresses are not affected by the used testing technique, the corresponding loading direction, or stress state. However, macroscopic samples (~1850 MPa) show slightly decreased stress values (Figure 2a and c). These differences can be explained by a slightly different grain size (Figure 1b) due to the imposed radius-dependent strain in the HPT device. The micron-sized pillars were FIB-milled at r ~14 mm, while macroscopic samples were machined from r ~12 mm to r ~14 mm. The larger grain size at the inner radius leads to an estimated flow stress decrease of ~250 MPa based on a simple Hall-Petch estimation [65,66], which is in accordance with the observed stress difference. Moreover, the elastic stiffness of the macroscopic compression tests is lower than in the pillar-compression tests since bottom and top faces of the samples deviate by <1° from perfect parallelism. The stress offset observable in Figure 2c is caused by slight pre-loading. However, due to the low to negligible work hardening, this is of minor concern for the data analysis at 8% flow stress.

(II) To check whether the flow stress decrease at elevated temperature results from an annealing effect during heating of the compression device or is induced by plastic deformation, a thermal annealing approach was conducted where bulk samples were annealed for 30 minutes at various temperatures. Grain heights (black) as well as grain lengths (red) are shown in Figure 6a, indicating a slight increase of grain size and a decrease of grain aspect ratio (blue) upon annealing. The initial average grain size of ~160 nm at r ~14 mm increased to ~238 nm at 200°C, ~294 nm at 300°C and ~315 nm at 400°C. The corresponding decrease in flow stress was calculated based on a Hall-Petch estimation [65,66] and results in ~200 MPa, ~300 MPa and ~350 MPa, respectively. Due to the generally larger mean grain size in the macroscopic ufg samples, the flow stress at 8% plastic strain decreased by 300 MPa to 1550 MPa at 200°C (Figure 2c). Consequently, reduced flow stress values at elevated temperatures mainly result from grain growth effects during heating the material in the compression device.

(III) The different strain hardening behaviour of the uniaxially deformed samples tested at RT (Figure 2a) and elevated temperature (Figure 2b and c) could be explained by increased thermal activation at elevated temperature. In this case, screw dislocations cross-slip and therefore lead to a reduced pile-up of dislocations during deformation, thereby reducing work hardening at elevated temperature. At RT dislocations do pile-up and strain hardening takes place.

(IV) Stochastic events are observed for sxx pillars at every temperature tested. During compression, the number of load drops depends on both the machine dynamics (strain-rate) and whether displacement or load controlled mode is used [67] and is an indication of discrete dislocation activity. For large ufg pillars, no intermittent flow is observable regardless of the used strain-rate. Here, burst events in individual grains are averaged out due to the high number of grains in the deformed sample volume. Serrations in the stress-strain curve of the deformed macroscopic sample at RT (Figure 2c) indicate failure events, which are also evident on the surface of the deformed sample (Figure 4a). At elevated temperature, such cracking did not occur for any used strain-rate, which is attributed to the higher dislocation mobility that helps reducing dislocation pile-ups causing local stress concentrations. The fact that cracking is not observed in pillars is attributed to the significant smaller number of large defects in the miniaturized sample volume [68].

Overall, the SRJ tests as well as constant strain-rate deformation under uniaxial as well as multiaxial conditions revealed good comparability across all tested length-scales and temperatures.

4.2 Surface to volume considerations

Globally, the flow stress is strongly dependent on microstructure, temperature and strain-rate, as shown in Figures 2 and 3. On a local scale, surface contributions, enhanced at small scales by an increase in surface-to-volume fraction, might cause differences in

deformation behaviour and should also depend on the ratio of sample dimension to grain size [22-25]. In Figure 4 and Figure 5, the local surface appearances are compared for different testing techniques. The detail in Figure 4b shows surface regions of ufg pillars at 230°C, where grains were pushed out of the sample surface due to high local deformation. Comparing the surface evolution with RT data, the temperature has no pronounced influence on the appearance of the surface. Considering the sxx case in Figure 4c, thermal activation causes a more localized deformation at elevated temperature [38], which is attributed to a more fcc-like slip behaviour.

Changing the applied stress state from uniaxial loading to multiaxial loading in nanoindentation, comparable observations were made on the residual impressions in ufg and sxx samples, as shown in Figure 5. In both cases of Cube Corner indentation, a pile-up formation with distinct differences in their appearance is observed. While slip lines are observed on the sxx sample surface in Figure 5b, grains partly emerge from the plastically deformed surface region in the ufg case. The residual surface topology increases with the amount of imposed strain (Berkovich 8%, Cube Corner 20%). No pronounced temperature effects were observed, as confirmed by an exemplary Berkovich impression performed at 300°C (Fig. 5a). Comparison with uniaxial in-situ pillar-compression experiments at RT and 230°C, at the same amount of plastic strain (Figure 5c and d), indicates almost no emergence of grains at ~8% global plastic strain. However, after ~20% strain (Figure 5c and d) several grains in the deforming areas were pushed out of the sample surface. This implies that the amount of applied strain is the dominant factor controlling surface topology evolution, while the stress state seems to be of minor influence. Investigations of the pillar volume, as shown in Figure 6b, reveal differences at individual temperatures. In Figure 6b, images I and II show the microstructure at RT and after annealing at 230°C, respectively. Grain growth occurred during thermal setup and sample alignment before mechanical loading. The grain aspect ratio was reduced according to Figure 6a. After compression, FIB cross sections of deformed pillars (Figure 6b,III and IV) reveal the compression induced reduction of grain height in highly deformed zones, indicated

by red arrows. In undeformed zones at the pillar base, the grain aspect ratio remained constant as shown for the undeformed state in Figure 6b,I and II. It is interesting to note that GB sliding [11,12] seems not to be dominant, as more sliding would be expected for higher deformation temperatures. This observation will be addressed in more detail in the next chapter.

4.3 Thermally activated deformation processes —from global to local flow behaviour

To examine the underlying deformation mechanisms in more detail, strain-rate sensitivities, m, were calculated [69] for all testing techniques using

m = dm (1)

at a representative strain of 8%. Results are presented in Figure 7a, where m of sxx and ufg Cr are plotted against Ttest. The obtained results are in good agreement with results on sxx [6] and cg Cr [1]. At RT, m for sxx (red) and ufg (blue) samples are around 0.07 and 0.02, respectively. The reduced m for ufg samples compared to the sxx state is attributed to the higher athermal stress component, caused by dislocations interacting with GBs in the ufg states [6,7].

For sxx Cr, increasing Ttest leads to a continuous decrease of m in Figure 7a. Above Tc (grey-shaded area), m-values in the order of 10_i are determined, indicating strain-rate insensitive fcc-like plasticity, where deformation is not governed by thermal activation of screw dislocations anymore. This behaviour is in good accordance with other studies [28,38,44], showing that overcoming Tc the thermally activated component diminishes.

In ufg samples, the general trend is significantly different from the sxx state. For temperatures below Tc, m slightly decreases due to increasing thermal activation, while interaction with GBs remains mostly unaffected. Overcoming Tc, m increases continuously with temperature. According to [6], this is mainly attributed to dislocation interactions with grain boundaries, as described also for ufg fcc metals [18,70]. An additional contribution to the strain-rate sensitivity could stem from the reduction of the athermal stress contribution due to slight grain coarsening (see Figure 6). Thus, the overall m is a mixture of both, thermally activated

dislocation-GB interactions and thermally activated grain coarsening. Notably, while the first contribution leads to an increasing m, the second one acts against it.

For further insights and to consider the influence of applied stress states especially in the sxx state, the corresponding activation volumes, v [71], at Ttest were calculated using

C • kB • Ttest „ „

v =-, (2)

m • a ' v '

where kB is the Boltzmann constant, and C a factor dependent on the microstructure. For ufg Cr, C was set to V3 based on the van Mises relation linking shear stress to normal stress in polycrystalline aggregates. For sxx samples, the factor is valid for nanoindentation, since a multiaxial stress state is present. For uniaxial compression testing on sxx pillars, C was set to the inverse Schmid factor (1/0.45), which corresponds to deformation on the expected {110} <111> slip systems [28,42]. To compare values of v, normalisation with the cubed Burger's vector b (2.5 • 10-10 m for Cr [72]) is common practice. Results are presented in Figure 7b, where v for sxx (red) and ufg samples (blue) are plotted against Ttest, respectively. For sxx samples at RT, v is ~9 b3, and a slightly increased value of ~14 b3 is observed for ufg samples. This is attributed to dislocation segments involved in formation of a double kink to overcome the next Peierls potential [8,43]. Comparable v for uniaxial bulk data have been already reported by Glebovsky and Brunner [4], Wu et al. [8], Wei et al. [33], Kim et al. [41] and Schneider et al. [43] for sxx W, nc Cr, ufg Fe and sxx Mo, respectively. Increasing thermal activation at higher temperatures leads to an increase of v to ~30 b3 for both microstructures until Tc is reached. Overcoming Tc, v still increases to values around 300 b3for sxx samples, indicative of a dislocation forest interaction process. In ufg samples, a rather constant v of ~30 b3 is observed for testing temperatures close to or exceeding Tc, independent of stress state and length scale. It is argued that dislocation-GB interactions dominate the deformation. The overall constant v from 100°C to 400°C obtained by several testing techniques over various length scales further demonstrates that the deformation mechanism does not change, even though grain coarsening occurred [73-75]. In other words, despite the vastly different surface-to-volume ratios probed

by the different experiments, different length scales and stress states, the observed deformation mechanisms are governed mainly by dislocation-based plasticity. The rather constant values of m and v with increasing strain, as shown in Figure 8a and b, further underline that the governing deformation mechanism does not change upon deformation, although with increasing amount of strain the dislocation density within the samples might vary, as evidenced from the differences in temperature dependent strain hardening behaviour as discussed in section 4.1. In literature, the initial dislocation densities of sxx and ufg bcc metals were reported to be ~1010 m-2 and ~1014 to ~1015 m-2 [76], respectively. Comparable values of 9.6 • 1014 m-2 and 3.5 • 1014 m-2 were reported for HPT deformed Nb and Ta, respectively [77]. Those differences in dislocation density might affect the flow stress values resulting in individual strength scaling behaviour in ufg compared to sxx samples, as reported in [78]. In the single crystal situation, the deformation behaviour is altered by substructure formation, as shown in pre-strained Ni samples [55] or during microstructural refinement by straining of copper crystals [79]. However, cell structures in the size range of 300 nm to 500 nm were reported. Indeed, in 160 nm sized grains, substructure formation is limited and GB's act as sinks and sources for dislocations [8]. Only few mobile dislocations are expected within the grain interior. Therefore, the accumulation of dislocations in the grain interior has only a minor influence on the governing deformation mechanisms, especially at higher strain values, as shown in Figure 8.

The standard deviations of m and v are small for ufg samples but larger for sxx samples. As stochastic dislocation behaviour is more pronounced in the sxx case, load drops mainly contribute to this scatter. For the present as well as previous results at RT [78], no indication of diffusional deformation processes was found, since in that case one would expect m-values of 0.006 to 0.009 and v-values of ~1b3 as reported for 30 nm sized bcc Fe [80] and 50 nm sized bcc Ta [81], respectively.

Several attempts have been made to interpret the varying deformation mechanisms in fcc and bcc metals with respect to a varying grain size. The basic fcc model was established by Conrad [18] and considers a change in activation volume with grain size

11 M2•G-b 1

* = ^ + ~ Wd12' ^ (3)

' ■ / ■ 1

where M = 2.9 is the Taylor factor for Cr [8], G = 116 GPa is the shear modulus, a = 0.36, KH-P is the Hall-Petch coefficient of 1380 MPam1/2 for Cr [82], v* is the activation volume related to thermal activation (see equation (2)) and vath was set as a constant activation volume related to athermal dislocation emission at a GB at constant temperature. v* varies with temperature, as the lattice friction stress (~50 MPa for Cr at RT [82,83]) diminishes with increasing thermal activation. The Conrad model was extended by Wu et al. [8] to estimate apparent m- and v-values for polycrystalline aggregates, using

Vatfc = $-d-b2, (4)

where £ is a grain shape coefficient which is constant for a certain temperature. Except equation (4), no further modifications of the fcc model were applied, as Conrad [18] already implemented a thermal stress component within the model to derive thermal activation volumes (v*). Results obtained by Wu et al. [8] show that m-values of Cr and Fe at RT can be correctly predicted over a wide range of varying grain sizes, as both, the friction stress (thermal) and dislocation interaction with GB's (athermal) contribute to the overall deformation. Moreover, the same requirements as mentioned for fcc metals hold true in bcc metals: the presence of intragranular dislocations restricted to their glide planes, dislocation pile-ups, and the absence of pronounced dislocation cells.

Values for £ in dependence of the grain shape are included in Figure 6a. In the present case, £ is temperature-dependent, as the aspect ratio of initially elongated grains decreased with increasing Ttest, which was observed during the annealing approach (see section 4.1 and Figure 6a). Figure 7 presents Conrad's model [18] (grey dashed lines) for the sxx as well as the

ufg case and the extended model of Wu et al. [8] (red dashed lines for sxx and blue dashed lines for the ufg case). Grey lines and the grey dashed area represent the minima and maxima of m-and v-values due to varying grain shapes. While both models fit for the sxx case, Conrad's model neglects grain coarsening and the change of the grain aspect ratio during thermal setup. It therefore underestimates m at increasing temperatures for the ufg material. Modifying the model of Wu by taking into account not only the changing grain size, but also changing grain shape coefficients ranging from 0.025 (RT) to 0.003 (400°C) (see Figure 6), the model fits the increasing m-values and low activation volumes at elevated temperatures well.

4.4 Strain-rate sensitivity vs. interface fraction

In Figure 9, the impact of different surface-to-volume ratios and number of involved interfaces on the determined strain-rate sensitivity is shown. Here, the number of grains across the plastic volume (magenta dashed line) and the fraction of grains which are affected by the sample surface (green dashed line) were estimated for different sample geometries during uniaxial and multiaxial testing. Therefore, grains were estimated to be of cylindrical shape with an aspect ratio of 3:1, typical for HPT deformation [36,78]. For rectangular-shaped samples an equivalent cylindrical sample diameter L was calculated. The number of grains contained in the plastic volume were estimated by calculating the sample volume and division by the before mentioned cylindrical-shaped grain volume. The fraction of grains which are affected by the sample surface were estimated using the specimen's surface subtracted by the top and bottom faces which are in direct contact with the flat punch indenter and the bulk material. This lateral area was divided by the average cross-sectional area of a single grain. Considering the different indentations, a simplified hemispheric plastic zone [84] after penetration to 2500 nm was taken into account for estimating the number of deformed grains per volume. The volume of the residual imprint was therefore subtracted from the hemispheric plastic zone. The remaining volume was further divided by the volume of a single grain, as mentioned before. To consider

the surface connectivity of grains, the base area of the hemispheric plastic zone was subtracted by the triangle-shaped surface area of the residual imprints and divided by the average cross-sectional area of a single grain. Based on this, a comparison of m-values (black), number of interfaces across the plastic volume (magenta), and fraction of surface-connected grains (green) is presented in Figure 9 as a function of the ratio between grain size vs. sample size d/L. The error bars indicate the resultant error which was derived by taking the standard deviation upon grain size determination into account (d=160 ± 51 nm). Also shown are representative stressstrain curves of different micro-pillars.

The light blue area in Figure 9 indicates the macroscopic regime, where the sample size is much larger than the grain size. Only a diminishing fraction of grains is located directly at the sample surface, and no influence of near-surface grains is observed during deformation. Overcoming a d/L ratio of ~0.02, the amount of surface-affected grains increases drastically. In this regime, indicated by the grey shaded area, grains noticeably emerge from the sample surface and might affect the deformation behaviour. Overcoming d/L values of ~0.1, ~50% of the grains touch the surface, and the grain size approaches the size of the plastic zone. For these states an sxx-like deformation behaviour is expected (yellow area). Notably, due to pillar aspect ratios of 3:1 and the used representative grain diameter in this simplified model, a fully sxx sample volume is statistically reached for d/L ratios larger than 1.33. Below this, individual GBs might affect the plastic behaviour [85]. However, as long as crystallographic slip traces reveal no intersections with GBs or other internal obstacles [78], dislocations can glide through the crystal and exit on the pillar surface, corresponding to slip events in sxx pillars. Such characteristics are evident in representative stress-strain curves of small ufg pillars (Figure 9, II-IV), where serrations and load drops are commonly observed. For macroscopic experiments, a smooth flow behaviour (Figure 9, I) due to the large number of grains in the sample volume and m-values of ~0.02 are observed. Increasing the fraction of surface grains leads to a slight decrease of m to ~0.014 and serrated flow arises in the stress-strain curves (Figure 9, II and III).

This is explained by slip events in individual grains. The scatter within evaluated m-values for such pillar sizes is quite large compared to bulk or sxx samples, because deformation is strongly affected by the local microstructure and crystal orientation. Figure 9, image III shows a 250 nm ufg pillar deformed in a uniaxial SRJ test [14,86]. Due to the stochastic deformation, the strain-rate jumps are hard to visualize and therefore marked with vertical dashed lines. Further increasing d/L ratios leads to a decreased scatter regarding m, as the probability for dislocations interacting with individual GBs is decreased, until m of sxx bulk samples and corresponding stress-strain curves (Figure 9 IV) are obtained.

5. Summary & Conclusion

The effect of temperature, stress state and strain-rate on the mechanical behaviour of ultrafine-grained (ufg) Cr was examined, spanning from the sub-micron regime by uniaxial pillar-compression and multiaxial advanced nanoindentation testing to macroscopic uniaxial compression experiments. Results in terms of temperature-dependent flow behaviour, strain-rate sensitivity and corresponding activation volume agree well within the different experimental techniques, demonstrating that length scales and stress states have only minor influence on the overall deformation. The main conclusions can be summarized as followed:

(I) Flow characteristics for room temperature deformation of ufg Cr agree well within the different techniques. The decrease of the flow stress above Tc mainly results from temperature induced grain coarsening during heating of the compression device. The decrease of the thermal stress contribution is of minor importance.

(II) Deformation of single-crystalline samples is dominated by the Peierls potential up to the critical temperature Tc. Overcoming this temperature, the thermally activated component to the flow stress diminishes and a strain-rate insensitive behaviour with further increasing activation volumes is observed.

(III) Up to ~0.87 • Tc, ufg samples deform by the thermally activated motion of screw dislocations, where the thermal stress contribution decreases with increasing temperature. This leads to a slight reduction of the strain-rate sensitivity.

(IV) Exceeding Tc, constant activation volumes and increasing m-values for ufg Cr are indicative for dislocation-grain boundary interactions as the dominant deformation mechanism. Grain coarsening due to annealing and a change in grain aspect ratio lead to a slight decrease of the activation volume, especially at elevated temperatures.

(V) The amount of implied strain during deformation shows significantly more influence on emerging surface grains than the stress state or the deformation temperature. The strain rate sensitivity and the activation volume remain constant with increasing strain indicating no changes in deformation mechanism. Grain boundary sliding has not been observed.

(VI) By extending existing models to incorporate the evolution of grain size and grain shape, the deformation mechanism characteristics m and u in ufg Cr have been successfully modelled for varying temperatures.

(VII) A transition in m is observed from a polycrystalline behaviour to an sxx situation, which is controlled by the amount of interfaces in the tested volume. Free surfaces and stress states are of minor concern.

With these novel insights, we hope to contribute to a better understanding of the size-dependent interplay between grain boundaries and sample dimensions and their influence on the deformation of bcc metals over several length scales. For future investigations, it would be of interest to discriminate the individual contributions of dislocation- grain boundary interaction and grain growth to the total strain-rate sensitivity.

Acknowledgements

The authors thank Franz Hubner and Robin Neubauer for sample preparation, DI Peter Kutlesa for help with the HPT device, Gabi Felber for preparation of the thinned lamellas, and Dr. Wolfram Knabl from Plansee for providing the material. Financial support by the Austrian Science Fund FWF (project number: P25325-N20) is gratefully acknowledged. Further financial support by the Austrian Federal Government (837900) (in particular from the Bundesministerium für Verkehr, Innovation und Technologie and the Bundesministerium für Wirtschaft, Familie und Jugend) represented by Österreichische Forschungsförderungsgesellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftförderungsgesellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme (MPPE, project A7.19) is appreciated.

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Table Captions

Table 1: Purity and bulk hardness obtained by nanoindentation and Vickers micro-hardness measurements

Sample Purity [%] Method Hardness [GPa]

sxx Cr 99.999 Nanoindentation 16 [6]

as-received polycrystalline Cr 99.9 Vickers 1.2 ± 0.04

ufg Cr, r ~14 mm 99.9 Vickers 4.4 ± 0.10

ufg Cr, r ~12 mm 99.9 Vickers 4.2 ± 0.10

Figure captions

Figure 1: Half of a HPT-deformed Cr disk with corresponding hardness map and BSE images of the microstructure. a) EDM was used to cut a lamella (black - left side) from the HPT disk for FIB pillar preparation from a disk radius of 14 mm and several macroscopic compression samples from a radius range between 12 mm and 14 mm. The compression axis is the axial direction of the HPT sample. b) Microstructures before and after HPT deformation. The left image shows the as-received Cr (d~200 |im). I and II represent the microstructure after HPT deformation at a disk radius of 14 mm (lamella for pillar preparation, d~160 nm) and at a disk radius of ~12 mm (for macroscopic samples, d~300 nm), respectively. For colour interpretation, the reader is referred to the online version.

Figure 2: Comparison of stress-strain curves at different strain-rates and temperatures for sxx and ufg Cr samples. a) 2 |im Cr pillars tested at RT, b) 4 |im Cr pillars deformed at 230°C. c) Macroscopic ufg samples at varying temperatures for a strain-rate of 3 • 10-3 s-1. For colour interpretation, the reader is referred to the online version.

Figure 3: Nanoindentation data of sxx and ufg Cr at variable strain rates and temperatures. a) Comparison between constant strain rate (CSM) and strain rate jump (SRJ) experiments. b) Hardness and Young' Modulus vs. indentation depth. Validity of the measurement is indicated by the constant Young's Modulus of ~303 GPa over indentation depth. c) Hardness divided by a constraint factor of 2.8 vs. indentation depth for sxx and ufg samples tested with Berkovich and Cube Corner indenter tips at varying temperatures. For colour interpretation, the reader is referred to the online version.

Figure 4: Post-deformation SEM images of macroscopic samples and micro-pillars. a) Macroscopic samples loaded at varying strain-rate and temperature. b) Ufg pillars with diameters of ~3 |im and ~4 |im tested at RT and 230°C, respectively. c) Deformed ~4 |im and ~6 |im sxx pillars at RT and 230°C. The insets indicate details of the deformed sample surfaces.

Figure 5: Residual impressions in a) ufg and b) sxx Cr performed with Berkovich or Cube Corner tips at varying temperature. For comparison, snapshots during in-situ SEM pillar-compression tests of ufg Cr at ~8% and ~20% plastic strain are shown in c) at RT and d) at 230°C. For details, see text.

Figure 6: a) Results of annealing experiments of the as-deformed material, corresponding decrease of grain aspect ratio and selected grain shape coefficients b) The microstructure of ufg pillars in the undeformed state at RT and 230°C (I,II) and the resulting microstructures after pillar compression (III, IV). For colour interpretation, the reader is referred to the online

version.

a 0 SO 100 150 200 250 300 350 400 450 0.1 -

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300 350 400 450 500 550 600 650 700

300 350 400 450 500 550 600 650 700

® Pillar Compression o Nanoiridentation <100'sxx 95 |jm Cr, Marcirikowski and Lipsitt[1] • Maier ef al. [61 - Wu ef al. [8], sxx Model » Pillar Compression , O Nanoindentation ° <> Maier ei al. [6] ■ Macro Compression Conrad ef al. [17], constant grain size ---Wu ef al. [8], ufg Model (extended: grain growth and grain shape)

Figure 7: Evolution of the (a) strain-rate sensitivity m and (b) activation volume v of sxx (red) as well as ufg (blue) samples with temperature, supported by model predictions of Conrad [18] (constant vath) and a modified model of Wu et al. [8] (vath = f(T), additionally accounting for

variable grain shape coefficients). The grey-shaded area indicates the critical temperature Tc of Cr. Some data [6] are taken with permission of Elsevier. For colour interpretation, the reader is referred to the online version.

Figure 8: Invariance of (a) strain-rate sensitivity m and (b) activation volume u with respect to

strain for individual uniaxial tests at variable temperatures for sxx and ufg Cr. The larger scatter in sxx values originates from stochastic dislocation behaviour. For colour interpretation, the reader is referred to the online version.

Figure 9: Strain-rate sensitivity (black), number of grains in the plastic volume (magenta) and corresponding connectivity of surface grains (green), dependent on sample size-normalized characteristic dimension d/L. The grey-shaded area shows a transition zone where grains tend to emerge from the sample surface. I) - IV) Corresponding stress-strain curves of pillars indicated. See text for more details. For colour interpretation, the reader is referred to the online

version.

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