2022 Volume 63 Issue 1 Pages 1-6
It has been established experimentally from the beginning of 21st century that the yield strengths at room temperature of submicron sized face-centered-cubic and body-centered-cubic metallic single crystals fabricated with the focused ion beam milling process increase dramatically with decreasing specimen diameter. In this paper, the mechanisms of the strengthening have been discussed theoretically on the basis of the single-ended source activation controlled deformation in submicron sized crystals. The experimental results of the power-law dependent yield strength with the decrease of specimen size have been reproduced quantitatively for both types of metallic crystals, where in some body-centered-cubic metals we take into account the effect of the Peierls potential for the screw dislocation motion on the source activation stress.
log(τc/G) vs. log(D/b) plot of critical dislocation bow-out stress component for six BCC metals. The thick line is the average value for each D/b calculated theoretically on the basis of the single-ended source activation controlled deformation in submicron sized crystals. Between the thick and thin dashed lines 95% of the strength distribution is included for each D/b value in the calculated distributions.
In the middle of 1950s, Brenner showed that the strength of grown metallic whiskers with the diameter from 1 to 15 µm increases with decrease of their diameter towards ideal strength of the crystals.1) Such anomalous increase of the strength has been interpreted by the lack of dislocations and of dislocation multiplication sources. More recently, by use of a high temperature molecular beam epitaxy method, Richer et al. succeeded in producing Cu whiskers with submicron diameter, which are absent of dislocations, showed strengths close to the ideal strength.2) On the other hand, thanks to the development of micromachining technique of focused ion beam (FIB) milling, tiny test samples with diameters from a few micrometers down to a few hundred nanometers became available. Since the pioneering research by Uchic, Dimiduk, Florando and Nix3) on the compression experiments in 2004 on micropillar samples of Ni and Ni3Al–Ta single crystals with the diameter from 0.5 to a few tens of micrometers, produced by the FIB milling process, mechanical tests on FIB processed submicron sized specimens became quite popular: from 2005 to 2020, the target crystals were mainly face-centered-cubic (FCC) single crystals of Au,4) Ni,5) Cu6) and Al,7) and from 2007 to 2019, body-centered-cubic (BCC) single crystals of Mo,8,9) W, Mo and Nb,10) W, Mo, Ta and Nb,11) V,12) Fe13,14) and Fe, Nb, and V,15) commonly with the ⟨100⟩ axial orientation, and also hexagonal-close-packed metals of Ti16) and Mg17) and some intermetallic compounds. An important difference between grown whisker crystals and nano-sized crystals fabricated by the FIB milling process is that in contrast to almost dislocation free whisker wires, FIB milling processed micro-pillars contain rather high density of dislocations of the order of 1013 m−2 18) which are most probably introduced as the damage of FIB milling.19)
The common features of the plasticity of submicron-sized single crystals are: (1) stress-strain curves are not continuous but stepwise consisting of a series of burst slips between the steps, indicating that the deformation proceeds by successive activation of multiplication sources. (2) the critical shear yield stress τc dramatically increases with decreasing specimen size D obeying a power law relation, though the data are considerably scattered, given by
| \begin{equation} \tau_{\text{c}}/G = A(D/b)^{m}. \end{equation} | (1) |
Theoretical calculation of the critical bow-out stress of single-ended dislocation multiplication source (abbreviated as single-ended source hereafter) has been performed by Pichaud et al.24) The result of the three dimensional discrete dislocation dynamics (3D-DDD) simulation of the strength of the single-ended source by Rao et al.25) gave essentially the same result as that by Pichaud et al. Tang et al.26) performed for the first time 3D-DDD simulations for the deformation of model FCC micropillar specimen containing randomly distributed Frank-Read dislocation sources, and concluded that increasing strength with decreasing size is due to lowering of mobile dislocation density. A larger scale 3D-DDD simulations have been conducted by Rao et al. for FCC nickel micropillars.27) They succeeded in reproducing typical discrete stress-strain curves and also in reproducing the power law relations of eq. (1) between the size and the critical resolved shear stress for the first time. They argued that among various mechanisms of hardening with decreasing size, the source hardening and exhaustion hardening are sufficient to explain the experimental results. However, eq. (1) was not theoretically justified. Parthasarathy et al.28) have performed for the first time a statistical analysis of the deformation of small sized crystal having randomly distributing pins of multiplication source on the slip plane. Assuming the longest arm of single-ended source or the weakest source controls the critical shear stress, they predicted size dependent strength and compared with experiments. They expressed the critical yield stress of micropillar crystals by additive expression of the three terms: friction stress, critical bow-out stress of the source and hardening term due to matrix dislocations. However, they used as the critical bow-out stress not for single-ended source but for double-ended source, leading to ≈ −1.0 for the size exponent of the strength, not reproducing experimental exponent. The same three terms expression has also been used by Ng and Ngan,29) and Norfleet et al.30) Among them, only Norfleet et al. used the correct expression for single-ended source activation stress. Based on their experiments on Ni micropillars of size dependent strength and observed dislocation structures, they assessed the validity of the theory and concluded that the theory cannot fully account for the strength of smaller microcrystals. In spite of these efforts of understanding the size dependent strength, universal relation of eq. (1) had not been reproduced.
In these situations, present authors31) recently succeeded in interpreting the power law size dependence of the strength of micropillars of FCC crystals quantitatively in terms of the theory of the single-ended source controlled deformation process. In the present paper, we try to interpret quantitatively the single-ended-source controlled deformation of submicron sized BCC crystals, as well as FCC crystals.
Most experiments so far made on BCC metal micropillars have been performed for single crystals with the ⟨100⟩ compression axis, except for a few studies with other compression axes.32–34) Brickman et al.8) performed nanopillar experiments for BCC Mo ⟨100⟩ single crystals in comparison with FCC Cu with the same orientation. They showed fundamental differences of the deformation behavior between them in the work hardening behavior and in the power law exponent of the size dependent strength. The present authors consider the former difference originates in the compression axis of ⟨100⟩ orientation, which is the stable orientation for compression with quadruple multiple slips in BCC crystal but unstable orientation leading to single slip in FCC crystal after some strain. The first systematic experiments for micropillar specimens of four BCC crystals were reported by Schneider et al.10) and found an interesting tendency, i.e., absolute value of the power law exponent m decreases with increasing Tc or the critical temperature at which the temperature dependence of the critical resolved shear stress (CRSS), controlled by the Peierls mechanism for screw dislocations, disappears. Kim and Greer11) performed tension and compression experiments for submicron sized ⟨100⟩ single crystals of Cu and Mo, and found tension-compression asymmetry of the strength in Mo, reflecting the asymmetry of the Peierls potential for twinning and anti-twinning shear in BCC crystals. Lee and Nix35) interpreted the results both for FCC and BCC metals by the three terms expression of Parthasarathy et al.28) by substituting the friction term with the critical shear stress of corresponding bulk crystal. They were successful in explaining the size dependence of both FCC and BCC crystals for $D \gtrsim 1$ µm. The problem is the use of incorrect critical bow-out term as mentioned above. Han et al.36) phenomenologically discussed the size dependent strength of BCC metals affected by the Peierls potential, based on the additive contributions from Peierls potential term and other strengthening term which is power-law dependent on the specimen size. They interpreted the change of the size dependent flow stress with the increase of the Peierls potential term. Rogne and Thaulow examined size dependent yield strength of iron micropillars,13) and interpreted the size dependent flow stress by dividing the specimen size into three regions: bulk region for over 10 µm size, truncated-source controlled region from 100 nm to 10 µm size and near theoretical strength region below 100 nm size. Huang et al.14) examined strain-rate dependence of the yield stress of iron micropillars with ⟨100⟩ orientation, and found that it is reduced by one order when the pillar size was reduced from 1000 nm to 100 nm, and that the exponent of the power law size-dependent strengthening is also strain-rate dependent. They attempted to rationalize the results by assuming that the flow stress consists of additive contributions of the friction term and the bow-out term of forest cutting process, both of which are strain-rate dependent. Yilmaz et al.15) selected three BCC crystals, Fe, Nb and V, of which Tc is commonly about 300 K, and investigated size dependent strength of their micropillar specimens at 296 K and at 193 K, where strength was defined by 4% flow stress. At room temperature, the size dependent initial flow stresses for three BCC crystals are almost the same as those of FCC crystals with the size dependent exponent of around −0.6, while at 193 K the exponent is around −0.3. Similarity of the room temperature behavior of the above three BCC crystals with those of FCC crystals is due to the fact that in both FCC and BCC crystals the mobilities of dislocations of both edge and screw characters are almost the same due to little contribution of the Peierls potential at room temperature. On the other hand, at low temperature the effect of the Peierls potential on screw dislocation motion makes the behavior similar to those in W and Mo in which the room temperature strength is determined by the Peierls potential for screw dislocations.
Thus, contributions so far made to the strength of micropillar specimens of BCC crystals already clarified important characteristics of BCC metals different from FCC metals. What is lacking at this stage is the quantitative understanding. The purpose of the present paper is to understand quantitatively the behavior of BCC micropillars by extending the previous single-ended source controlled model applied to FCC micropillars.31)
The yield stress of bulk single crystals is believed to be determined by the resistance to dislocation glide without taking care of multiplication stress of the dislocation source, in spite of the fact that the yielding is always accompanied by dislocation source activation. This is based on the fact that due to the inhomogeneous distribution of matrix dislocations producing hard and soft regions, dislocation multiplication is done in soft regions under lower stress than the dislocation glide resistance in hard regions. Considering that the size of the dislocation cells formed in metallic crystals after weak deformation is 1∼2 µm, the wave length of the inhomogeneous distributions of dislocations with the average density of ρ ≈ 1013 m−2 will be of the order of 1 µm. Thus, the dislocation distribution in micropillars with diameter of the order of 1 µm or less will be more or less homogeneous and the yielding of micropillar metallic crystals should be multiplication-source activation controlled.
In pure FCC crystals in which the Peierls potential is negligibly small due to the dissociation of the dislocations into Shockley partial dislocations, the critical stress of bow-out of the single-ended source with the length L is given by
| \begin{equation} \tau_{\text{y}}(L) = (\beta Gb/L) \ln (L/b) + \tau_{\rho}. \end{equation} | (2) |
log(CRSS/G) vs. log(D/b) plot of the data points for FCC metal micropillars collected by Shahbeyk et al.23) The thick line is the average value for each D/b theoretically calculated for round specimen.31) Thick dashed-line indicates the minimum value and between the thick and thin-dashed lines 95% of the strength distribution is included for each D/b value in the calculated distributions.
For BCC crystals with the ⟨100⟩ orientation, log(τc/G) vs. log(D/b) plots are more variable than those of FCC crystals. The distribution of the data points of the critical shear stress for all the BCC crystals collected by Shahbeyk et al.23) expands over a wide band region with a negative slope. The width of the distribution is not much different from that of FCC metals, but what is different is that the distribution of the data points of BCC metals is more or less uniform in the band, in contrast to the rapidly decreasing point density away from the theoretical curve in FCC metals. Even for the same BCC crystal, the data points are considerably scattered, possibly originated from the compression axis deviation around ⟨100⟩ orientation; depending on the degree of deviation and its direction from the exact ⟨100⟩ orientation, the deformation starts either by single slip or multiple (double, triple or quadruple) slips. As a result, the yield stresses of single crystals near ⟨100⟩ orientation tend to scatter from crystal to crystal depending sensitively on the way of deviation from the exact ⟨100⟩ axis. However, since ⟨100⟩ orientation is the stable orientation for compression for BCC metal, after some strain hardening all the samples deform by quadruple slips keeping a large work hardening rate. Thus, to evaluate the validity of the single-ended source controlled deformation model, we selected experimental data of the initial flow stress at several % strain, instead of the critical yield stress. Here, we confirmed that the data scatter in these experiments is relatively small, comparing with that of the critical yield stress or the flow stresses at 0.2–1%. Those selected experimental data are W,10) Mo,10) Ta,11) V,12) Nb10) and the data by Yilmaz et al.15) for Fe, Nb and V, which are plotted in Fig. 2. By comparing the data point distribution of FCC metals, those of BCC metals are located at higher region. Two reasons are considered: effect of the Peierls potential for screw dislocation motion and the work hardening due to multiple slip. Due to the thermally activated glide of screw dislocations, the yield stress of BCC metals decreases with increasing temperature, and the Peierls potential controlled component τPeierls disappears above a critical temperature Tc which depends on the height of the Peierls potential. Tc values for high-purity BCC metals, estimated from CRSS vs. temperature curve for ultra-high purity single crystals are 770 K for W,37) 470 K for Mo,38) 380 K for Ta39) and around 300 K for Fe,40) Nb41) and V42) for the conventional strain-rate of 10−3–10−4 s−1. Since we are interested in the room temperature deformation, τPeierls component appears only in those metals with Tc higher than 300 K, i.e. W, Mo and Ta. Comparing the data of FCC metals in Fig. 1, the overall data scattering of more or less systematic data for six BCC metals in Fig. 2 is not much different. As already clarified in the pioneering research on BCC metals by Schneider et al.,10) absolute value of the exponent m becomes smaller as Tc becomes higher. Also as already shown by Yilmaz et al.15) whose results are included in Fig. 2, for the results for BCC crystals with Tc value at about room temperature, the size dependence of the strength is almost the same as those of FCC crystals, which has been interpreted due to the common feature of high mobility of dislocations for both edge and screw characters without affected by the Peierls potential for screw dislocation.
Multiplication sources in BCC micropillar crystals are considered to be produced by the double-cross slip mechanism of screw dislocations. Double-ended multiplication sources may be produced. However, screw dislocation glide at room temperature in any BCC metal is more or less wavy by nature due to frequent cross-slips.43) As a result, after whirling around two poles of a double-ended source leading to escape a part of it out of the surface, the source dislocations initially connected to two poles have little chance to meet and return to the initial source position again, and the double-ended source eventually changes into two single-ended sources. Ryu et al.44) showed in their DD simulation for BCC micropillar that specimen surface induced cross slip of ⟨100⟩ screw dislocation gliding on a slip plane making an acute angle with the specimen surface can produce a single-ended source by cross-slip. In any case, the character of the single ended sources will be near screw dislocation.
In the bow-out process from the near screw single-ended source dislocation, kink-pair formation and kink migration are always accompanied, as depicted in Fig. 3. Bow-out process toward the critical state from near screw single-ended sources proceeds either by successive formation of a kink at the surface, which is regarded as truncated kink-pair taking the image force of the kink into account, followed by rapid migration of the kink (case (a) and (b)) or successive kink-pair formation on screw dislocation followed by rapid kinks migration (case (c)). Both the kink-formation at the surface and kink-pair formation inside the surface are thermally activated process and the stress needed for these processes will be denoted as τkp hereafter. If we approximate τkp for screw dislocation motion is constant as mentioned below, τkp can be regarded as a friction stress for screw dislocation motion. Note that the portion of the bowing source dislocation with high density of kinks will change to smooth non-screw dislocation by collapsing the kinks, and can migrate without kink-pair formation, but we would stress that during the course of bow-out process τkp stress is always necessary for the source dislocation near ⟨100⟩ orientation. As the bow-out process proceeds, the curvature of the source dislocation increases toward the critical bow-out state as depicted in Fig. 2. The exact estimation of the backward stress acting on the source dislocation due to the bow-out at the critical state is not simple, but may be approximated by the result for FCC metal given by Pichaud et al.,24) and we assume that the critical backward stress given by the first term of the right side of eq. (2) acts on every part of the bow-out source dislocation. Thus, in pure BCC metals, the activation stress of the single-ended source near screw orientation with the length L consists of three components, i.e., backward stress due to the critical bow-out, the resistance due to interaction with matrix dislocations and the kink-pair formation stress acting on screw part of the bow-out dislocation, written as
| \begin{equation} \tau_{\text{y}}(L) = (\beta Gb/L)\ln (L/b) + \tau_{\rho} + \tau_{\text{kp}}. \end{equation} | (3) |
Schematic drawing showing the bow-out process of single-ended sources with near screw orientation in BCC crystal, which proceeds by successive kink formation at the surface and kink migration in (a) and (b) or successive kink-pair formation on screw dislocation and kinks migration in (c).
Assuming the initial dislocation density of 1013 m−2, dislocation density after a few % compression along ⟨100⟩ direction will be of the order of 1014 m−2. Using the Taylor relation, τρ is estimated to be ∼1.5 × 10−3G. We subtracted τρ and τkp values for three BCC crystals W, Mo and Ta from the initial flow stress in the previous subsection, and we obtained the size dependence of the bow-out stress component τbowout (first term of right-hand side of eq. (3)) for the three crystals. Those data are respectively plotted in Fig. 4, and approximate m values for each crystal before and after the subtraction are given in Table 1. m values for τbowout for the three crystals are found to be around −0.6, almost consistent with the theoretical value of about 0.66.31) For other crystals, we subtracted τρ component, and we plotted in Fig. 5 τbowout components for six BCC crystals together with the theoretical average value drawn by the thick line. We find that the distribution of τbowout components for BCC metals is almost the same as that of FCC crystals in Fig. 1, which strongly indicates that deformation of BCC micropillars is also controlled by the single-ended source activation.
log(τy/G) vs. log(D/b) plot of three BCC metals of W (a), Mo (b) and Ta (c). Filled circles are observed values and open circles are the bow-out term (first term of right-hand side of eq. (3)) for each specimen obtained by subtracting τρ and τkp terms from each observed τy value. Approximate exponent m for each data is listed in the attached table.
In the past two decades, strengths of submicron-sized single crystals have been intensively investigated mainly for FCC and BCC metals. However, the mechanism of the power law dependent increase of the strength with decreasing specimen size had not been clarified quantitatively. In the previous paper,31) the present authors have shown that the size dependent yield strength of micropillar FCC crystals can be interpreted quantitatively by the single-ended source activation controlled deformation mechanism based on the theoretical result of activation stress of single-ended source. In the present paper, experimental data for BCC metals with the ⟨100⟩ axial orientation can also reasonably be interpreted by the single-ended source controlled deformation by taking the Peierls potential term and the matrix hardening term into account.
