Effect of annealing on nanoindentation slips in a bulk metallic glass

We measure and analyze the statistics of nanoindentation pop-ins in a bulk metallic glass that has been annealed after being subjected to high pressure torsion. We argue that these pop-ins are avalanches of slip events. Larger slip avalanches are observed after annealing at higher temperatures. The slip avalanche size distribution shows scaling behavior and suggests the existence of a critical annealing temperature. Results are shown to be consistent with the predictions of a simple mean field theory. This implies that the statistics of slip avalanches in nanoindentation can be used to extract information about the structure, history, density, and free volume of the material. The methods employed here are expected to be applicable to a wide range of materials.

Figure 1

Geometry of high-pressure torsion experiment. A disk sample (10 mm diameter and 0.85 mm thickness) is placed between two cylindrical anvils. A compressive load (typically 5 GPa) is applied and the lower anvil is rotated at a constant speed.

Figure 2

(a) X-ray diffraction pattern and (b) high-resolution transmission electron microscopy image with corresponding selected area diffraction pattern from $\mathrm{Z}{\mathrm{r}}_{50}\mathrm{C}{\mathrm{u}}_{40}\mathrm{A}{\mathrm{l}}_{10}$ BMG deformed by HPT with 50 revolutions.

Figure 3

(a) Schematic illustration of annealing treatments and polishing and nanoindentation tests. (b) DSC heating curves for each heating cycle. There is an exothermic peak corresponding to structural relaxation when the peak temperatures range from 150 to 400 °C, and the onset temperature of relaxation rises after each heating run. The increase of onset temperature in each curve reveals that the amount of free volume is dependent on the annealing temperature. The sample was heated up to 600 °C to see the glass transition and crystallization peaks.

Figure 4

Optical microscopy image of the cross section of the sample deformed with 50 revolutions. The red square indicates the $36\times 36\mu {\mathrm{m}}^2$ region for the nanoindentation tests.

Figure 5

Typical load-depth curves of the as-deformed $\mathrm{Z}{\mathrm{r}}_{50}\mathrm{C}{\mathrm{u}}_{40}\mathrm{A}{\mathrm{l}}_{10}$ BMG, and after annealing at 200 and 400 °C. The inset shows scanning probe microscope images of indents.

Figure 6

Illustration of interaction between the indenter tip and the surface of the sample. The region immediately surrounding the indenter tip is the hydrostatic core, the region between $R_1$ and $R_2$ is the plastic region, and beyond $R_2$ is the elastic region [26]. The slip avalanches are expected to occur in the plastic region.

Figure 7

Complementary cumulative distribution functions (CCDFs) for the slip avalanche sizes observed in all the trials using the first method of analysis, plotted for six annealing temperatures. Forty-nine trials were analyzed for each annealing temperature on a sample of $\mathrm{Z}{\mathrm{r}}_{50}\mathrm{C}{\mathrm{u}}_{40}\mathrm{A}{\mathrm{l}}_{10}$. The avalanche size is defined in the text. The CCDFs show that on average the avalanche sizes tend to increase for increasing annealing temperature.

Figure 8

Average value of slip avalanche durations as defined by the second method of analysis, binned by size. The flattening out observed for annealing temperatures $T=200{}^{\circ }\mathrm{C}$ and $T=250{}^{\circ }\mathrm{C}$ indicates a change in slip avalanche dynamics. The mean field predictions from [7, 8, 25] suggest that the slip avalanches in the flat regions are spanning slip avalanches rather than scaling slip avalanches. Thus we choose the size at which the curves flatten as the maximum slip size for those annealing temperatures. Consistent with the model predictions, we only obtained a scaling collapse of the CCDFs for slip avalanches smaller than those maximum sizes.

Figure 9

CCDFs for slip avalanche sizes after removing the largest slip avalanches. The inset shows a scaling collapse of the CCDFs. The collapse variables τ and θ are set to their mean field theory values, which are $\tau =\frac{3}{2}$ and $\theta =2$. The critical temperature was varied until the best collapse was found, which is at $T\mathrm{c}=500{}^{\circ }\mathrm{C}$. The glass transition temperature for this sample of $\mathrm{Z}{\mathrm{r}}_{50}\mathrm{C}{\mathrm{u}}_{40}\mathrm{A}{\mathrm{l}}_{10}$ is $424{}^{\circ }\mathrm{C}$, which is lower but means that a critical temperature of 500 °C is not unreasonable. As in Fig. 8, these are slip avalanches as defined by the second method of analysis.

Figure 10

Complementary cumulative distribution functions (CCDFs) of slip avalanche sizes using the first method. Each of the CCDFs incorporates data from 49 experimental trials. The data from a single nanoindentation run consists of the load on the indenter head as a function of time and the total displacement of the indenter head as a function of time. Both of these quantities were sampled at a rate of 200 Hz. In the analysis method used for this figure the load signal is divided by the square of the displacement signal in order to give a signal that is approximately proportional to the applied stress as a function of time. A sudden drop in stress, and thus a negative spike in its time derivative, signals a slip avalanche. But the variations in the time derivative of the stress were much larger for early times and much smaller for later times, so in order to find a signal that was roughly constant over the time of interest, the discretely differenced stress as a function of time was multiplied by the total displacement of the nanoindenter raised to the power of 1.4. When this signal dropped below the small negative threshold the start of the avalanche was marked, and when the signal came back above the threshold the end of the avalanche was marked. The size of the avalanche is defined as the change in stress from the beginning to the end of the avalanche multiplied by the total displacement at the beginning of the avalanche raised to the power of 1.4.

Figure 11

CCDFs of slip avalanche sizes using the second method. This figure shows CCDFs of avalanche slip sizes taken from the same data as Fig. 10. But in the method of analysis used for this figure the displacement signal is discretely differenced to get a signal showing the derivative of the displacement with respect to time. A sudden rise in the displacement signals a slip avalanche, so positive spikes in the time derivative of the displacement should signal a slip avalanche. But the displacement signal has an overall trend that scales roughly like the square root of the time, thus the derivative has a trend that scales like the reciprocal of the square root. So a least-squares fit was done for the derivative signal and the resulting fit was subtracted off, leaving only the fluctuations in the derivative signal about the local mean. When this signal went above the small positive threshold the start of the avalanche was marked, and when the signal came back below the threshold the end of the avalanche was marked. The size of the avalanche is defined as the change in displacement from the beginning to the end of the avalanche.

Figure 12

CCDFs of slip avalanche sizes in TPa $\mathrm{n}{\mathrm{m}}^{1.4}$ created using the first analysis method for annealing temperatures $T=200{}^{\circ }\mathrm{C}, T=250{}^{\circ }\mathrm{C}$, and $T=300{}^{\circ }\mathrm{C}$. Each plot contains slip avalanches from a specific part of the trial: beginning, middle, or end. The slip avalanches were measured in the loading segment of the indentation, which was 20 s long. The windows divide the loading segment into thirds, so the CCDF for the first time window shows only those slip avalanches that occurred during the first third of the loading segment of a trial. Likewise the second time window shows those slip avalanches that occurred during the middle third, and the third time window shows those slip avalanches that occurred during the final third. These plots indicate that the slip avalanche sizes are not strongly correlated with what time during the loading segment at which they occur.

Figure 13

CCDFs of the slip avalanche size taken from indentations for three different annealing temperatures. The data are also separated according to when during the trial the avalanche occurred. The time windows correspond to those defined for Fig. 12. Each graph shows two CCDFs: one in which the slip avalanche size is calculated using the first method and the other in which the slip avalanche size is calculated using the second method. To compare the methods directly, the $x$ axis in the second method was rescaled so that both CCDFs started at the same size. Hence the slip avalanche size is given in the units of the first method, TPa $\mathrm{n}{\mathrm{m}}^{1.4}$. The CCDFs appear to be similar in all nine plots, suggesting that the two different methods for detecting and measuring avalanches give similar results.

Figure 14

CCDF of slip avalanche sizes as defined by method 1 without eliminating the smallest detections.

Figure 15

CCDF of slip avalanche sizes as defined by method 2 without eliminating the smallest detections.