Statistical Approach for Understanding the Effect of Specimen Size on the Yield Stress and Its Scattering in Mechanically-Alloyed Cu and ODS-Cu Obtained by Micro-Pillar Compression Test

Abstract

Oxide Dispersion Strengthening copper (ODS-Cu) alloy has high strength, high thermal conductivity and superior irradiation resistance which are needed for fusion reactor divertor components. In this study, specimen size dependence of the yield stress in Cu and ODS-Cu which were made by mechanical alloying and hot pressing was investigated by micro-pillar compression test. The distributions of yield stress values were different between the sizes, 1, 3 and 5 µm cubic micro-pillars, which can be explained by the inhomogeneous spatial distributions of both the crystallographic grain in matrix and oxide-particles.

1. Introduction

Due to the extreme environment of fusion reactor divertors, the heat flux components required good thermal conductivity with the stability of properties at high temperatures. Neutron irradiation effect on their mechanical properties such as irradiation hardening is also an essential problem. Currently, oxide dispersion strengthened copper (ODS-Cu) alloy has been recognized as one of the promising candidates for divertor heat sink.¹⁾ The ODS-Cu alloy Cu–0.5 mass% Al₂O₃ (Glidcop Al25) which are made by an internal oxidation technique was considered for ITER divertor, showing a good irradiation resistance and high-temperature strength.²⁾ Recently, we successfully fabricated new ODS-Cu alloys with fine oxide particles (Y₂O₃) distribution at the average particle size of 5 nm by mechanical alloying (MA).³⁾ The ODS-Cu alloys had a good ultimate tensile strength (UTS) of 491 MPa with a good ductility of 19%.⁴⁾

Nanoindentation hardness tests have a long history to evaluate irradiation hardening of ion-irradiated materials.⁵⁾ Our recent study suggested that a bulk-equivalent hardness in the ion-irradiated subsurface can be derived from depth profiles of the ion-irradiated Fe-based ferritic alloys based on the indentation size effect, damage-gradient effect and softer substrate effect.⁶⁾ However, the relationships between the bulk-equivalent hardness and Vickers hardness and further tensile/compression yield stress is still unclear. Compared with nanoindentation hardness tests, ultra-small testing technologies (USTT) such as micro-pillar compression tests⁷⁾ and micro-tensile tests⁸⁾ has a great superiority to evaluate the yield stress directly from the samples fabricated from the ion-irradiated subsurface by focused ion beam (FIB). Among them the micro-pillar compression test is useful to examine a much number of specimens. In various USTTs, however, specimen size effect can significantly affect the mechanical properties such as yield stress and work hardening behavior. The very first study of specimen size effect for single crystal Fe, Cu and Ag was published in 1956 by Brenner.^9,10) With decreasing in the specimen size, the mechanical strength becomes higher accompanied by a larger scattering typically. To explain the Brenner’s experiment result, a statistical model was developed based on a random spatial distribution and orientation of dislocations, which can be used to predict the average yield strength and the scattered data for single crystal materials.¹¹⁾

When the micro-pillar compression test is applied for ion-irradiated materials, the obtained mechanical properties may be different with the bulk specimen due to specimen size effects. Therefore, this study investigates the micro-pillar compression yield stress and its distribution in pure Cu and ODS-Cu (Cu–1 mass% Y₂O₃) which were made by MA followed by hot press. Statistical analysis based on microstructural observation and strengthening mechanisms was examined to understand the effect of the specimen size on the value and scattering of yield stress evaluated by micro-pillar compression tests.

2. Experimental Procedure

This study used three kinds of Cu materials, single crystal-Cu (SC-Cu), mechanically-alloyed and sintered pure Cu (MAed-Cu) and ODS-Cu with 1% weight percent yttrium oxide (Y₂O₃). MAed-Cu and ODS-Cu were prepared by using high purity (99.9%) copper powder with 45 µm in particle size and high purity (99.99%) Y₂O₃ powder with 0.4 µm in particle size as raw materials. The powder was mixed and sealed into the milling pot in the glove box under Argon environment. The ratio between stainless-steel balls (Φ10 mm) to powder is 10:1. MA was carried out by using a high energy ball milling machine Emax (Retsch) under a water-cooling condition at a rotation speed of 500 rpm with a 30 min–5 min cycle, which means milling 30 min than stopped 5 min for cooling. The actual milling time with this cooling cycle was 24 hours. By using this method, the temperature during the MA process was well controlled at temperatures of pot surfaces below 26°C. After MA, the powder was consolidated under Ar environment by Hot Press (HP) at 800°C for 45 min with a load of 8.0 kN. The specimen size after hot press was a cylinder with 15 mm in diameter and 10 mm in height.

After fabrication, Vickers hardness test with a load of 0.2 kgf was performed on SC-Cu (from ⟨100⟩{100} orientation), MAed-Cu and ODS-Cu. The crystallographic grain structure of MAed-Cu and ODS-Cu were observed by electron backscatter diffraction (EBSD) in field emission-electron probe micro analyzer (FE-EPMA) JXA-8530F. EBSD data were analyzed by the OIM software. The EBSD sample preparation was performed by fine polishing with SiC paper up to 4000 grit and then with diamond spray up to 0.25 µm and finally with colloidal silica suspension. As for ODS-Cu, the oxide particles were observed by transmission electron microscopy (TEM) with scanning transmission electron microscope (STEM) and energy dispersive X-ray spectrometry (EDS). TEM samples were prepared by using focused ion beam (FIB) followed with fine mill by using Gentle Mill machine.

Micro-pillar compression tests were performed by Nanoindenter G200 equipped with a flat diamond punch tip. Three different sizes of micro-pillar (1, 3 and 5 µm’s cube) was fabricated on the surface of SC-Cu ({100} surface), MAed-Cu, and ODS-Cu by using FIB and examined to evaluate the compression yield stress.

3. Results

3.1 Vickers hardness test

Vickers hardness measurements were conducted to obtain the average mechanical properties for SC-Cu, MAed-Cu and ODS-Cu. Table 1 shows the result of Vickers hardness test. The hardness of ODS-Cu was much larger than MAed-Cu.

Table 1 Vickers hardness result.

3.2 EBSD

Figure 1(a)(b) shows the grain map of the EBSD result of MAed-Cu and ODS-Cu. It appears that the size distribution of grains in both MAed-Cu and ODS-Cu are not Gaussian ones as can be seen from the size distribution in Fig. 1(c). Therefore, two methods of calculate grain size was used in the present study, which is average grain size calculation and expected grain size calculation:   

\begin{equation} d_{\text{average}} = \frac{\sum \nolimits_{k = 1}^{n}d_{k}}{n} \end{equation} (1)

  

\begin{equation} d_{\text{expected}} = \sum \nolimits_{k = 1}^{n}d_{k} \times p_{k} = \sum \nolimits_{k = 1}^{n}d_{k} \times \frac{A_{k}}{A_{\textit{total}}} \end{equation} (2)

where n is the total number of grains, p is the area fraction of each grains, A_k is area for k_th grain, A_total is the total area of the measured area in EBSD test. Table 2 shows the average grain size and expected grain size of MAed-Cu and ODS-Cu derived from Fig. 1. The expected values of grain size, which were calculated from grain size distribution in area fraction as shown in Fig. 1(d), were much larger than the average grain size values.

Fig. 1

EBSD Mapping, (a) MAed-Cu, (b) ODS-Cu, (c) Grain distribution in number fraction, (d) Grain distribution in area fraction.

Table 2 Grain Size of MAed-Cu and ODS-Cu.

3.3 TEM

The microstructure of Y₂O₃ particles dispersed in the ODS-Cu was observed by TEM. Figure 2 shows the High-angle Annular Dark Field Scanning TEM (HADDF-STEM) and STEM-EDS result of the ODS-Cu. In the ODS-Cu, the Y₂O₃ can be observed in darker contrast than Cu matrix. The STEM-EDS images shows that the composition of dark contrast part in Fig. 2(a) contains Y and O.

Fig. 2

HADDF-STEM and EDS result of ODS-Cu, (a) HADDF, (b) EDS of Cu, (c) EDS of Y, (d) EDS of O.

Figure 3 shows a large area STEM-HADDF image of ODS-Cu. Electron energy loss spectroscopy (EELS) is used to measure the thickness of the TEM specimen in Fig. 3 which is used to calculate the volume of TEM specimen. In Fig. 3, eight areas divided by 1 µm square are selected to count the size and number of oxide particles. Combining with the volume of specimen calculated from the EELS result, the number density, volume fraction and size distribution of Y₂O₃ in each square was calculated as shown in Table 3.

Fig. 3

STEM-HADDF image of ODS-Cu.

Table 3 Counting result of Y₂O₃ in ODS-Cu.

Figure 4 shows the size distribution of oxide particles in the ODS-Cu. The most frequent grain size was around 8 nm. However, there also exist oxide larger than 80 nm and oxide smaller than 5 nm.

Fig. 4

Size distribution of oxide particles in Cu–1 mass%Y₂O₃.

3.4 Micro-pillar compression

As a comparison group to MAed-Cu and ODS-Cu, single crystal Cu’s micro-pillar compression data was included in this study. Not like MAed-Cu and ODS-Cu, SC-Cu has no grain boundaries and no oxide particles. During the micro-pillar compression test, most of the SC-Cu’s pillar shows barrel type deformation due to the friction at the contacting surface and multi-slip condition which is similar with most MAed-Cu and ODS-Cu’s micro-pillar compression test. To use other direction in compression test for SC-Cu with specific slip has advantages to study the CRSS of single crystal Cu. However as a comparison group, multi-slip condition with barrel type deformation is more reasonable.

Figure 5 shows the stress-strain curves of micro-pillar compression test for SC-Cu, MAed-Cu and ODS-Cu. Figure 6 shows the size dependence of compression yield stress. For MAed-Cu and ODS-Cu, the distributions of yield stress were different between 1, 3 and 5 µm cubic micro-pillars. Smaller pillar had a wider range of yield stress for ODS-Cu. For MAed-Cu, only one 5 µm size pillar shows much higher yield stress than others. SC-Cu’s data shows less-scattered yield stress for all size’s pillar. Despite the large distribution in the compression yield stress, it can be seen that the increase in the pillar size lead to decreases of the yield stress for MAed Cu and ODS-Cu. However, SC-Cu has a size-independent yield stress in the present study. D. Kiener has reported that yield stress of micro-pillar of SC-Cu became larger for <1 µm size specimen.¹²⁾

Fig. 5

Stress-strain curves of micro-pillar compression test, (a) SC-Cu, (b) MAed-Cu, (c) ODS-Cu.

Fig. 6

Yield stress of micro-pillars.

4. Discussions

If the micro-pillar compression test is applied for the present MAed-Cu and ODS-Cu after ion-irradiation, the large distribution of compression yield stress must lead to the difficulty to evaluate irradiation hardening. The reproducibility of compression yield stress in the SC-Cu with small deviations indicates that the large distribution of compression yield stress in MAed-Cu and ODS-Cu is not due to the experimental procedure such as fabrication and compression but due to the intrinsic inhomogeneous micro-scale mechanical properties. From the EBSD results, both MAed-Cu and ODS-Cu have inhomogeneous spatial distribution of grain size. TEM micrograph also shows inhomogeneous spatial distribution of oxide particles. Such microstructural features are responsible for strengthening these materials.

Previous researchers also showed that the ratio between grain size and specimen size has effect on specimen’s yield stress especially for small specimen.^13,14) For the present experiment, due to inhomogeneous spatial distribution of grain size, it is hard to determine the actual grain structure for each micro-pillar, which means it is more complicated to determine the specimen size to grain size ratio for smaller pillars. However, according to L.V. Raulea’s work,¹⁴⁾ for multigrain specimen, with increasing in the specimen size to grain size ratio, the mechanical strength becomes higher. However, in this study, 3 µm and 5 µm size’s pillar has yield strength at same level while 1 µm size pillar has highest yield stress value, which indicated this specimen size to grain size ratio effect has no impact in this study.

Therefore, we consider that the large distribution of yield stress in the micro-pillar compression tests could be due to the inhomogeneous microstructure such as grain size and oxide dispersion. We assumed that strengthening mechanism for MAed-Cu consists of grain refinement strengthening and dislocation strengthening but that of ODS-Cu can further contain oxide dispersion strengthening.

Firstly, inhomogeneous spatial grain distribution is considered for grain refinement strengthening by applying Hall-Petch relation.^15,16) As for Hall-Petch relation, many studies indicated that for copper, there is a yield stress decrease below a critical grain size from 10 to 100 nm.^17–25) In the present study, the grain size of both ODS-Cu and MAed Cu is much larger than 100 nm which indicated that hall-petch relationship is still valid to study grain refinement strengthening behavior in this study.

Since the micro-pillar size is at the same level of grain size in the present study, the average grain size could be different from pillar to pillar. Therefore, the grain refinement strengthening possibly has a certain distribution dependent on the pillar size. To determine the distribution of grain refinement strengthening, a program using Python was designed based on Monte Carlo method as shown in the flow chart of Fig. 7.

Fig. 7

Flow chart for Monte Carlo simulation of grain refinement strengthening.

The process of the program is described as follows:

1. 1)    Firstly, an area, 1 µm², 9 µm² and 25 µm² which is the same as the cross-section of micro pillars, is set.
2. 2)    Different size grains which are randomly selected from the area distribution of grains obtained by the EBSD result (shown in Fig. 1(d)) were used to fulfill the set area. This is repeated for 10000 times.
3. 3)    Average grain size of all grains used for each fulfill operation are returned.

By using the calculated average grain size from the Monte Carlo simulation, the distribution of grain refinement strengthening for each pillar can be obtained as shown in Fig. 8 by applying the Hall-Petch law:   

\begin{equation} \sigma _{y} = \sigma _{0} + K \times d^{ - 1/2} \end{equation} (3)

where σ_y is the yield stress, σ₀ is a materials constant, for Cu is 20 MPa,²⁶⁾ K is the strengthening coefficient (a constant specific to each material), for Cu is 0.14,²⁶⁾ and d is the average grain diameter. The simulation result implies that 1 µm size pillar has the largest distribution range of grain refinement strengthening, while 5 µm pillar has the smallest distribution range. The average strengthening value of 1 µm size pillar is higher than the other pillars. For MAed-Cu, the size effect of grain refinement strengthening still exists even in the 3 µm size pillar. For ODS-Cu, the average grain refinement value and strengthening distribution for 3 µm pillar and 5 µm pillar are at the same level, indicating that the pillar size larger than 3 µm can avoid the size effect in grain refinement strengthening for ODS-Cu.

Fig. 8

Distribution of grain refinement of different size pillars for MAed-Cu and ODS-Cu, (a) MAed-Cu, (b) ODS-Cu.

Secondary, inhomogeneous spatial distribution of oxide particles is discussed for dispersion strengthening. The dispersion strengthening by oxide particles is quantitatively estimated from the TEM data. The relationship between oxide distribution and oxide dispersion strengthening is given by the following equations:²⁷⁾   

\begin{equation} \frac{\sigma _{\text{or}}}{G}=\frac{AMb}{2\pi \lambda } \times \left[\textit{ln}\frac{D}{r_{0}} + B \right] \end{equation} (4)

A = 1/(1 − ν), B = 0.6 for screw dislocations   

\begin{equation} \lambda = 1.25l_{\text{s}} - 2r_{\text{s}} \end{equation} (5)

  

\begin{equation} l_{s} = \frac{\sqrt{2\pi \overline{r^{3}}} }{\sqrt{3f\bar{r}} }, \end{equation} (6)

  

\begin{equation} r_{s} = \frac{\pi }{4}\left(\frac{\overline{r^{2}}}{{\bar{r}}}\right) \end{equation} (7)

where G is shear modulus (44.7 GPa for Cu), ν is the Poisson ratio (0.34 for Cu), M is the Taylor factor (3.0),²⁸⁾ b is the magnitude of the Burgers vector (2.56 × 10⁻¹⁰ m).²⁹⁾ r₀ is the inner cut-off radius of a dislocation core, which assumed to be the same magnitude of Burgers vector. λ is the average face-to-face distance between particles on a slip plane, r_s is the average particle radius, l_s is the average center to center distance between particles, r is particle radius. By using the counting result of oxide particles in each area as shown in Table 3, the dispersion strengthening in ODS-Cu can be calculated as shown in Fig. 9. The estimation indicates that the dispersion strengthening value in the micro-pillar is also inhomogeneous in ODS-Cu and the difference between the largest and the smallest dispersion strengthening values is 94 MPa.

Fig. 9

Distribution of oxide dispersion strengthening.

Finally, the dislocation strengthening can be calculated by the following equation:³⁰⁾   

\begin{equation} \sigma _{\text{dis}} = aMGb\rho ^{1/2} \end{equation} (8)

where α is a constant 0.5,³¹⁾ M, G and b is same as the eq. (2), ρ is the dislocation density which can be estimate from XRD result of the sintered material by using the modified Williamson-Hall plot.³²⁾ The result of dislocation strengthening is shown in Table 4.

Table 4 Dislocation Strengthening in MAed-Cu and ODS-Cu.

Estimated yield stress values were deduced by a summation of dislocation strengthening, grain refinement strengthening, and oxide dispersion strengthening as calculated above. The comparison of estimated yield stress and experimental one is shown in Table 5. The dispersion range is determined by the difference between the maximum value to the minimum value for both estimated value and experimental value. Most experiment values fit the estimated values except 5 µm size MAed-Cu pillar which shows relatively high yield strength compared with other 5 µm size MAed-Cu data from Fig. 6. With a limited group of compression results, the reason for this is currently hard to explain. It is considered that to get the true distribution of yield stress for each size’s pillar, more experimental data point is required.

Table 5 Comparison of dispersed data between theoretical value and experimental value.

In the present study, most pillar shows barreling type deformation due to low aspect ratio, the friction at the contacting surface and multi-slip condition. However, in the view of using micropillar’s data to evaluate bulk material’s properties, barrel type deformation with multiple slip is better than shared type deformation which is more suitable to study the deformation mechanisms. Besides, for ion-irradiated specimen, the affected depth of ion-irradiation for specimen normally within 2 µm. And, due to the existence of heavy damaged zone, the actual depth can be use is just from 0 to 1000 nm. To keep the total micro pillar in this region with comparable number of grains inside micro pillar is the key issue in further experiment. Therefore, 1 µm size cubic micro-pillar is chosen as the smallest pillar for the present study. Based on the experiment result and the statistical analysis provided in this study, the effect of aspect ratio in yield stress and size effect is not severe compared with effect of inhomogeneous microstructure such as grain size and oxide dispersion.

Therefore, although the size effect and large distribution of compression yield stress exist in 1 µm size pillar’s compression test for both MAed-Cu and ODS-Cu, the range and distribution of yield stress is predictable with this estimation procedure. With enough data point, micro-pillar compression test can be used to evaluate irradiation hardening for ion-irradiated MAed-Cu and ODS-Cu.

5. Conclusion

In this study, yield stress of Cu and ODS-Cu with 1 mass%Y₂O₃ which were made by MA and hot-pressing were investigated by micro-pillar compression tests. The following conclusions were obtained:

1. (1)    The EBSD results showed that the grain size of ODS-Cu was finer than that of MAed-Cu and the grain size distributions of both materials were spatially inhomogeneous.
2. (2)    The spatial distribution of oxide particles was also inhomogeneous. While the average oxide particle size is around 8 nm, there also exist very large oxide particles with diameter more than 100 nm.
3. (3)    Micro-pillar compression tests using 1, 3, 5 µm cubic specimens showed that smaller pillar had wider range of yield stress. Statistical analysis of the possible three kinds of strengthening mechanism, such as dislocation strengthening, grain refinement strengthening, and dispersion strengthening, indicated that the inhomogeneous spatial distribution of grains and oxides could be responsible to the size dependence of range in yield stress obtained by the different size of micro-pillars.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers 16H02443 and 19H02643. And, the authors wish to thank Mr. Can Zhao of Institute for Materials Research, Tohoku University, for the cooperation in programing of the Monte Carlo simulation.

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