† Corresponding author. E-mail:
In situ synchrotron angle-dispersive x-ray diffraction experiments on niobium powders have been conducted at pressures up to 61 GPa and room temperature using the diamond anvil cell technique. From the full width at half maximum of the measured diffraction lines, the yield strength was derived with the line-width analysis theory. The niobium powder sample was found to be compressed more packed firstly and then yielded at ∼14 GPa–18 GPa. Following an initial increase in the yield strength with pressure, an obvious decrease was observed occurring at ∼42 GPa–47 GPa accompanying with a typical pressure dependence above 47 GPa. The experimentally observed anomalous softening of the yield strength in niobium surprisingly follows the trend of the predicted unusual softening in the shear modulus by the recent theoretical investigations. The possible mechanisms, applicable to interpret the yield strength softening of materials at high pressure, were also discussed in detail.
Recently, the group-VB transition metals (V, Nb, and Ta) have attracted numerous theoretical and experimental studies[1–10] due to their unusual mechanical properties associated with the peculiar electronic band structure effects under pressure. Theoretical studies[1–3] predicted a pronounced softening in the trigonal shear modulus
The shear modulus sets a natural scale for yield strength in the framework of the theory of dislocations,[11] so this extraordinary softening of the shear modulus of the VB metals might lead to an unusual effect on the yield strength and other mechanical properties at the relevant pressure, even an anomalous softening in the yield strength. The yield strength, the minimum stress that allows one to plastically strain a material, is of great interest partly because of the underlying physical connection with shear modulus. For that reason, the yield strength of the VB metals attracted numerous experimental investigations.[12–16] Klepeis et al.[12] observed a decrease in the yield strength of vanadium starting at ∼40 GPa–50 GPa both in polycrystalline foils and powders, following the softening trend of the shear modulus. Jing et al.[13] observed an obvious reduction in the yield strength of tantalum at ∼52 GPa–69 GPa with typical pressure dependence above 69 GPa, which is in accordance with the softening trend of the shear modulus by theoretical prediction[3] and IXS measurements.[10] Dewaele et al.[14] measured the yield strength of single crystal tantalum up to 93 GPa. In the two experimental runs, they observed a distinct yield strength decrease at above 51 GPa and 84 GPa, respectively. Considering the possible orientation difference between the single crystals used in the two experiment runs, the experimentally observed yield strength decrease still followed the trend of the shear modulus qualitatively.[3,10]
However, in vanadium and tantalum’s group VB relative niobium, the measured yield strength in previous experiments[15,16] did not show the expected yield strength softening under pressure. Using the radial XRD method[17] in combination with the lattice strain theory[18,19] and the line-width analysis method,[20–25] the yield strength of niobium was experimentally investigated up to 40 GPa.[15,16] The experiment results showed that the yield strength of niobium initially increases with pressure and follows a subtle decrease with a shallow maximum at ∼12 GPa–13 GPa, and then restored a monotonic increase beyond 13 GPa (the decrease in “yield strength” below 13 GPa may be in connection with the yielding of the initial sample, as discussed in Section 4). The discrepancy between the experimental observation[15,16] and the theoretical prediction[1–3] motivates us to revisit the yield strength of niobium at high pressures.
In this work, the yield strength of pure niobium powder is determined up to 61 GPa at room temperature using the diamond anvil cell (DAC) combined with synchrotron angle-dispersive x-ray diffraction. An obvious strength softening in niobium is observed. Our experiment provides data to validate theoretical models and to develop insights into the fundamental physical mechanisms controlling the pressure-dependent yield strength of niobium.
The stress state of polycrystalline sample, nonhydrostatically compressed between two diamond anvils of DACs, is generally considered as a superposition of a macroscopic differential stress and a microscopic deviatoric stress,[20] which result in macro- and micro-strains, respectively. The microscopic strain causes the broadening of the diffraction lines and the macroscopic one leads to the diffraction lines shifting. The broadening and shift of the diffraction lines is valuable information for the mechanical properties of the sample. The diffraction-line broadening, an indicator of the distribution of the longitudinal micro-strain parallel to the diffraction vector, is used to analyze and to extract the high-pressure yield strength. This method is termed as the line-width analysis method.[16,21–25] In the framework of the line-width analysis theory, the diffraction-line broadening due to microscopic deviatoric strain and crystallite size varies with diffraction angle under angle-dispersive x-ray diffraction geometry as[21,22]
![](cpb_27_10_106201/cpb_27_10_106201_ieqn2.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn3.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn4.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn5.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn6.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn7.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn8.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn9.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn10.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn11.gif)
For a cubic system,
![](cpb_27_10_106201/cpb_27_10_106201_ieqn12.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn13.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn14.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn15.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn16.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn17.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn18.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn19.gif)
![](cpb_27_10_106201/cpb_27_10_106201_ieqn20.gif)
![]() | Table 1.
The single-crystal elastic constants |
The experiment was performed with 300-
Typical results of fitting Eq. (
![]() | Fig. 2. (color online) Some typical results of fitting Eq. ( |
The microscopic deviatoric stress, calculated by the slope
![]() | Fig. 3. (color online) The microscopic deviatoric stress as a function of pressure. The experimental data measured in this work are denoted by the open diamonds. Those of Ref. [16] obtained from microscopic deviatoric stress analysis (S2015-1) and microscopic deviatoric strain analysis (S2015-2) are denoted by the solid squares and asterisks, respectively. Those of Ref. [15] (S2011) are denoted by the solid triangles. |
The good quality of our data unequivocally indicates that there are three unusual regimes in the microscopic deviatoric stress versus pressure plot as shown in Fig.
The second anomalous regime is between 30 GPa–35 GPa and the third one is between 42 GPa–47 GPa. The yield strength indicates a plateau between 30 GPa–35 GPa, and then follows a continuous decrease above 42 GPa, accompanied with a minimum at 47 GPa. The decrease in the yield strength is in accordance with the slopes of the fitting lines, as shown in Fig.
![]() | Fig. 4. (color online) Pressure versus volume data for niobium at 300 K. The solid circles are the experimental data measured under nonhydrostatic conditions in this work and the solid line is the 3rd order Birch–Murnaughan fit. The open up-triangles and right-triangles are the experimental data determined by Takemura et al.[36] in helium (Takemura–He,) and methanol–ethanol–water (Takemura–MEW) pressure media, respectively. No distinct discontinuity in pressure–volume was observed. |
Some researchers[39,40] advised that the texture/preferred orientation of the sample at high pressures could lead to the strain softening. The preferred orientation/texture could be indicated by a non-uniform distribution or changes in the intensity of the Debye–Scherrer diffraction rings.[12] Figure
![]() | Fig. 6. X-ray diffraction patterns of a niobium powder sample at 18, 35, 46, and 61 GPa on MAR345 image plate. |
Meade and Jeanloz[39] found the yield strength of MgO varies with the different initial thickness of the sample (thus different strain at high pressures) and attributed it to the preferred orientation of MgO. However, in one of their independent experiment runs, the yield strength of MgO monotonously increased with elevating pressures, which is essentially different from how niobium behaves in this work. Furthermore, the effect of the preferred orientation is a continuously smooth function of pressure,[41] and the yield strength-pressure curve would level out once the texture saturated,[42,43] which cannot interpret the observed decrease in yield strength of niobium. Additionally, the texture/preferred orientation in materials with bcc structure is not so remarkable as those with face centered cubic structure because of the reduction of slip systems, but the observed softening of the yield strength in niobium was more than 15%, an amount even significantly larger than the reported critical case of hcp-Co,[42,43] which is a highly anisotropic material. Weir et al.[40] not only observed similar strain softening in Ta samples with different initial thickness as that in MgO,[39] but a notable falling of yield strength in each independent experimental run. They attributed this yield strength anomaly to material damage induced in highly strained specimens. A finite element modeling[44] revealed that the continuous decrease of high-pressure yield strength of tantalum[40] could be caused by unexpected influence from the cupping of the diamond anvils. Nevertheless, Jing et al.[13] found that the yield strength of tantalum decreases at ∼52 GPa–69 GPa but followed a continuous increase from 69 GPa till the highest pressure 101 GPa, which can never be interpreted by material damage or cupping of diamond anvils. Obviously, the decrease of yield strength in niobium is quite different from the cases in MgO[39] and Ta.[40] Both strain softening and cupping of diamond anvils fail to explain the yield strength variation with pressure, especially the decrease in yield strength above 42 GPa along with a continuous increase.
In the Steinberg strength model,[45] the strain and pressure are the major factors to affect the high-pressure yield strength at room temperature. They both enhance the yield strength at relatively low pressures, whereas the strain hardening component is dominant and constrained by setting the yield strength to an upper limit. After saturation of the plastic strain, the pressure hardening component is the dominant hardening mechanism. The corresponding typical characteristics is that the yield strength versus pressure curve has an initially larger slope followed by a smaller one, such as in lithium fluoride.[46] In this work, the anomalous softening of yield strength in niobium is above 42 GPa, a pressure far beyond the plastic strain it saturates at, and pressure hardening is dominant. Therefore, it is not unreasonable to conclude that the anomalous softening of yield strength in niobium is pressure dependent and caused by the pressure softening of the shear modulus. A theoretical investigation suggested that the latter could be a combined effect of Fermi surface nesting, band John–Teller effects, and electronic topological transition in this pressure range.[38] Our further confidence to make such a conclusion is from the support of niobium’s group VB relatives, vanadium and tantalum. An unusual yield strength decrease under pressure was also observed in vanadium[12] and tantalum[13,14] and followed the trend of the predicted shear modulus softening by theoretical studies.[1–3] It should be noted that the theoretical calculation of shear modulus is on the basis of single crystal, whereas the experimental measurement is from the powder sample. A comparison between these results is in principle valid, but the differences should exist to a certain extent. That is why the experimentally observed yield strength softening of VB metals[12–14] qualitatively agree with the predicted shear modulus softening.[1–3,9] The analogous characteristics of yield strength softening in vanadium and tantalum and their consistence with the theoretical prediction make us more confident to believe that the pressure- dependent softening of yield strength in niobium has a close connection with the unusual behavior of the trigonal shear modulus.
In conclusion, the microscopic deviatoric stress of niobium has been experimentally investigated up to 61 GPa. Three distinct anomalous regimes were observed. Between 14 GPa–18 GPa, the microscopic deviatoric stress slightly decreased with pressure due to yielding of the packed niobium powder sample. Above 18 GPa, the microscopic deviatoric stress can denote the yield strength of the polycrystalline niobium. The second anomalous regime is from 32 GPa to 35 GPa, where the yield strength levels out. The strongest softening is observed from 42 GPa to 47 GPa, where the yield strength evidently decreases with pressure. The comprehensive analysis shows that the yield strength softening in niobium is independent of phase transition, strain softening caused by preferred orientation (along with material damage), and the cupping of diamond anvils. The experimentally determined pressure-dependent yield strength of niobium behaves similarly to the theoretically predicted pressure-dependent trigonal shear modulus. We argue that there is a systematic relationship between the anomalous strength softening of niobium and the unusual softening in the shear modulus. Further experimental evidence from quantitative texture/preferred orientation analysis and IXS measurement will be helpful to evaluate the strain effect and to determine the pressure dependence of the shear modulus in the concerned pressure range, respectively.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] | |
[46] |