Orientation-dependent indentation response of helium-implanted tungsten

A literature review of studies investigating the topography of nano-indents in ion-implanted materials reveals seemingly inconsistent observations, with report of both pile-up and sink-in. This may be due to the crystallographic orientation of the measured sample point, which is often not considered when evaluating implantation-induced changes in the deformation response. Here we explore the orientation dependence of spherical nano-indentation in pure and helium-implanted tungsten, considering grains with<001>,<110>and<111>out-of-plane orientations. Atomic force microscopy (AFM) of indents in unimplanted tungsten shows little orientation dependence. However, in the implanted material a much larger, more localised pile-up is observed for<001>grains than for<110>and<111>orientations. Based on the observations for<001>grains, we hypothesise that a large initial hardening due to helium-induced defects is followed by localised defect removal and subsequent strain softening. A crystal plasticity finite element model of the indentation process, formulated based on this hypothesis, accurately reproduces the experimentally-observed orientation-dependence of indent morphology. The results suggest that the mechanism governing the interaction of helium-induced defects with glide dislocations is orientation independent. Rather, differences in pile-up morphology are due to the relative orientations of the crystal slip systems, sample surface and spherical indenter. This highlights the importance of accounting for crystallographic orientation when probing the deformation behaviour of ion-implanted materials using nano-indentation.

Four 500 nm deep spherical nano-indents were made for each orientation for both implanted and unimplanted areas (MTS NanoXp, Synton ~4.2 µm radius diamond tip, 50 µm spacing between indents). Use of a spherical indenter tip removes the additional complexity of in-plane indenter orientation relative to the crystal associated with non-rotationally symmetric tips, e.g. Berkovich. SEM micrographs (Zeiss Merlin FEG SEM) of the nano-indents in <111>, <001> and <110> grains in the helium-implanted and unimplanted material are shown in Appendix C. They clearly show that consistent results are obtained across the indents for each crystal orientation.
To quantify pile-up morphology, AFM was carried out on one indent per grain orientation in the implanted and unimplanted material ( Figure 1). AFM measurements were done in contact mode using a Digital Instruments Dimension 3100 AFM with Bruker CONTV-A tips (10 nm nominal tip radius). Little difference is seen between indents in different grains of the unimplanted sample (Figure 1 (d)-(f)). However, in the implanted samples there are striking orientation dependent variation in pile-up morphology (Figure 1 (j)-(l)). The <001> grain shows the characteristic large pile-up and distinct slip steps we previously observed [22].  Orientation-dependent differences are also seen in the load-displacement curves shown in Figure 2. Each curve is the average the load-displacement response recorded from the four indents per orientation. In the unimplanted grains there is little difference between the different crystal orientations. In the helium-implanted material, on the other hand, the load for <001> orientation is ~20% higher than for <011> and ~30% higher than for <111>. To explore the origin of the orientation-dependence of indentation behaviour in the heliumimplanted material we consider a CPFE model of the indentation process. Recently we developed a CPFE formulation to simulate nano-indentation of a helium-implanted <001> oriented tungsten single crystal [28]. The formulation is based on a hypothesis derived from a comparative study of nano-indentation and micro-beam Laue diffraction measurements performed on helium-implanted and unimplanted parts of a tungsten <001> crystal [22]. Here, we use the same CPFE formulation with the appropriate orientation matrix to simulate indentation of the <011> and <111> grain of the polycrystalline tungsten sample. All other parameters of the model were kept constant. Below we briefly discuss the model and the underlying hypothesis and further details can be found elsewhere [28]. Micro-beam Laue diffraction measurements of the deformation zone beneath indents in a <001> single crystal, showed a more tightly confined plastic zone in the helium-implanted sample than in the unimplanted material [22]. Increased pile-up, slip steps and increased indentation load were observed in the implanted material. Based on these observations, we hypothesised the following mechanism for the interaction of glide dislocations and heliumimplantation-induced defects (known to consist predominantly of Frenkel pairs that cannot recombine as helium occupies the vacancy [2]). We propose that initially, helium-defects strongly obstruct gliding dislocations and cause a pronounced hardening. However, with progressive deformation, passing dislocations facilitate release of helium from the defect cluster and consequently recombination of Frenkel pairs. Reduced defect density channels are thus formed that allow easier propagation of subsequent dislocations. This leads to a localisation of deformation, which in turn is the origin of the large pile-up and slip steps observed for the <001> orientation. A model based on this hypothesis was implemented in a CPFE user material subroutine (UMAT) for Abaqus where strain softening was applied to the helium-implanted layer. The UMAT is based on a user-element developed by Dunne et al. [29] and is founded on the theory of multiplicative decomposition of the deformation gradient into elastic and plastic components [30][31]. Briefly, the CPFE formulation constrains slip to applicable slip-systems (assumed to be the 12 {110} slip planes with a/2<111> slip vector directions [32]). The slip rate is governed by a physically-based constitutive law that considers the glide of thermally activated dislocations in a field of pinning obstacles [29]. Taylor hardening is implemented, where the critically resolved shear stress (CRSS), with initial value 4 5 , is increased as a function of evolving densities of geometrically necessary dislocations (GNDs) [33]. The CRSS in the unimplanted sample is 4 = 4 5 + < = ?@A . A modified form of this equation is used in the helium-layer; 4 = 4 5 + < = ?@A + B where the additional B term accounts for the increased resistance to dislocation glide due to helium-defects. To account for the strain softening, B is reduced at the end of each time increment, ∆ , if the material point was deformed plastically. The reduction in B represents the gradual formation of defect-free regions, and consequently easier dislocation glide. B is considered to be a function of the total accumulated crystallographic slip and the rate at which it decreases is estimated to be proportional to its current value i.e. the current helium-defect concentration b . This suggests an exponential softening: where ̇P W is the crystallographic slip rate on slip system , P E and P EQ∆E are the total accumulated crystallographic slip, summed over all slip systems, at the start and end of the time increment, and B 5 is the initial value of B . Only three UMAT parameters were fitted to the nano-indentation and AFM results of the <001> grain: the initial unimplanted CRSS, 4 5 , the hardening prefactor and only one additional parameter for the helium-layer, the softeningrate γ. All other UMAT parameters, including B 5 were physically-derived, or taken from literature values [28] . The geometry for CPFE simulations was a 3D, 20×20×20 µm 3 sample block and a 4.2 µm radius spherical indenter simulated in Abaqus (Dassault Systèmes, Providence, RI, USA). Based on symmetry, for the <001> and <011> grains, the model simulated one quarter of the experimental setup. For the <111> grain, a model simulating a third of the experimental setup was used. The 20 µm high block was partitioned into two layers: a 3 µm thick top layer and 17 µm thick bottom layer. When simulating indentation on the helium-implanted tungsten, the top layer was assigned material parameters of implanted tungsten and the bottom layer that of pure tungsten. The indenter was subjected to a displacement of 0.5 µm into the sample block. A structured finite element biased mesh with >39500 20-noded quadratic hexahedral elements with reduced integration was used (C3D20R) (Appendix D); with an element size of 50 nm at the indent. The simulated indent surface-profiles, after unloading, for both samples are shown in Figure 1. The AFM micrographs in Figure 1 are rotated to have the same in-plane orientation as the profiles predicted by CPFE. The simulated load-displacement curves for each grain orientation in both the implanted and unimplanted samples are shown superimposed on the experimental measurements in Figure 2. A good quantitative agreement is observed between CPFE and experimental results for the unimplanted sample, particularly evident in the load-displacement curves. Four, two and three-fold symmetry can be seen in the CPFE predicted surface profiles of the (001), (011) and (111) grains respectively (Figure 1 (a)-(c)).
The CPFE model reproduces the experimental pile-up for all three grains in the implanted sample remarkably well. In particular, the model captures the much lower pile-up in the <011> and the <111> grains compared to the <001> grain. In terms of the mechanical response, the <001> grain reaches ~14% higher load than the <011> and <111> grains; with the latter two producing a very similar load response. It is important to note here that the parameters of the model were determined solely based on the indentation results of the <100> oriented crystal. For the simulations of the <011> and <111> oriented grains all parameters except for the input crystal orientation were kept unchanged. Anisotropy in indentation behaviour has been previously noted in ∝-Ti [35] and Be [36] polycrystals, where modulus and hardness decreased significantly with increasing angle of inclination between the c-axis and the indentation axis. Orientation-dependent mechanical performance has been seen in tungsten too, where the <001> single crystal was found to be the best performing kinetic energy penetrator, owing to favourable slip during loading and shear localization [37]. It is interesting that orientation-dependent differences in indentation response are as pronounced at the nano-scale. The results highlight the importance of determining the crystallographic orientation of the ion-implanted sample for accurate evaluation of the ioninduced alteration in deformation behaviour. Quantitative agreement between CPFE predictions and experimental results for the implanted sample inspires some confidence in the hypothesis that localised deformation through slip channels, formed by dislocations interacting with helium-defects, can cause the large pile-up and increased hardening. In the 1960s' Makin et al. proposed a mathematical theory linking the creation of large pile-up and slip-steps to an accelerating fall in the force resisting dislocation glide [38]. As the defect reduction rate will depend on the number of defects, an exponential decrease in B as implemented in the CPFE formulation in Eq. (2), is reasonable. The model uniquely captures the four-fold pile-up increase in the <001> grain in the implanted samples, affirming the strain-softening hypothesis. Strain-softening and consequent slip channel formation in irradiated materials, both fcc and bcc, has been experimentally observed in numerous studies [39]- [41]. This is of particular concern as it may lead to untimely failure due to loss of ductility. The CPFE predictions also provide further insight into the underlying deformation zone beneath indents. Comparison of the field of effective plastic strain beneath indents in each grain for both samples (Figure 3) shows an orientation-independent trend. The deformation field becomes more confined near the indent in all grains in the implanted material, compared to a more widespread deformation zone in the unimplanted material. Put together, these results, suggest that the underlying mechanism responsible for the modified behaviour of the implanted material, i.e. the interaction of dislocation with helium-induced defects, is orientation-independent. The changes in the pile-up morphology for different crystal orientations are simply a product of the relative orientations of the active slip systems, the indenter and the sample surface.
In summary, we have examined the effect of helium-implantation in grains of three different orientations. It was found that the (001)-oriented grains showed ~70% increased hardness and ~172% increased pile-up compared to unimplanted tungsten. In contrast, the (011) and (111) grains show negligible change in pile-up and only a ~30% increase in hardness compared to the unimplanted sample. CPFE based on the application of strain-softening in the heliumimplanted layer, was able to reproduce the experimental results for all three grains with surprising accuracy. The fact that significantly different indentation behaviour is observed for different grain orientations in the implanted material highlights the importance of considering crystal orientation when interpreting nano-indentation data.

Details of the CPFE formulation
Details about the formulation can be found elsewhere [28]. The values for the material properties used in the formulation is provided below in Table D.1.

Applied Boundary Conditions
The boundary conditions applied to the indentation model in Abaqus included symmetric XZ and YZ planes, a traction free top surface, and fixed displacement and rotation boundary conditions on the remaining surfaces. An additional boundary condition was applied to the helium-implanted layer to account for helium-implantation induced residual stresses. Detailed description of the method of calculation of residual stresses generated by helium-implantation can be found elsewhere [1]. Briefly, the residual stresses can be expressed as a function of the out-of-plane lattice swelling i.e. oo induced by helium In a prior study, on a 001-single crystal tungsten implanted with helium under similar conditions, oo tuv , the deviatoric component of the out-of-plane strain, was measured by whitebeam Laue diffraction [22]. Here oo is estimated from this prior obtained oo tuv ; oo = 3 2 oo tuv ⁄ . It is assumed that ff and ii component of the total strain tensor is zero, (deformation along X and Y directions are restricted in order to maintain geometrical continuity between the implanted layer and the substrate). Knowing oo tuv in the helium layer to be ~ 530 × 10 -6 , ff p = ii p are computed to be -260 MPa. 2. Scaling with effective modulus The sample block was assigned elastic properties of tungsten i.e. elastic modulus of 410 GPa and Poisson's ration of 0.28 [42]- [44]. The indenter, designed as a discrete rigid wire frame was not assigned any material properties. This was done to avoid a full meshing and to allow increased simulation size. The material properties of the diamond indenter tip (with modulus of w =1143 GPa) used in the experiment, was accounted for in the simulation by scaling the results with an effective modulus Eeff (322.58 GPa), where,