Influence of Displacement and Load Control on Nanoindentation Mapping Results

Mechanical microscopy using nanoindentation mapping is a powerful technique for materials characterization, providing highly spatially resolved mechanical properties data from microstructures. However, different control modes are possible for performing nanoindentation maps, primarily load or displacement control to set load or displacement targets. In this application note, we examine the influence of these control modes on the mapping results on an Al—Cu eutectic sample featuring two very mechanically distinct phases. Displacement control is shown to give more accurate and precise results due to the more consistent contact depths achieved in both phases. This arises from two factors: significantly higher measured stiffness values in the harder phase, due to higher attained forces required to achieve the target depth, and consistent indentation size in the softer phase preventing unwanted interactions with neighboring indentations.

Introduction

Mechanical microscopy is the application of high-speed nanoindentation to characterization of the spatial variation of mechanical properties within a microstructure. As an indentation consists of a plastically deformed impression into a sample’s surface, the question of resolution immediately arises as to how closely indentations should be spaced together in a map to achieve the maximum resolution for a given indentation size.

Traditionally, Samuels and Mulhearn’s criterion [2] requires that indentations should be spaced at least three times the width of the indentation, or 20× the normalized spacing vs. indentation depth (d/h) for a Berkovich indenter. This is sufficient to prevent any contact between neighboring indentation plastic zones. Phani and Oliver [3] reinvestigated this criterion using 3D finite element modeling and experimental validation. They determined that a significantly closer spacing (10 d/h or 1.5× the width of the indent) was achievable without significant interference for most homogeneous materials.

However, homogeneous materials are not the normal subject of indentation mapping, as the reason for performing an indentation map is to characterize the spatial variation in mechanical properties within a complicated microstructure. Therefore, the presence of dissimilar phases requires that variation in mechanical behavior is expected, so how can a consistent indentation size and spacing be achieved? In load-controlled operation, the resulting indentation size is a direct function of the hardness of the indented region, so to ensure all indentations meet the desired spacing criterion the applied load must be carefully selected so that the biggest indentations in the softest regions still satisfy the criterion. This automatically implies that the indentations in the harder phases will be smaller than necessary to satisfy the criterion. In displacement-controlled operation, all indentations are made to the same maximum penetration depth, so that the final indentation size is controlled by the elastic unloading, rather than the plastic deformation.

In previous application notes, several benefits of displacement control for indentation mapping have been discussed: consistent indentation size for ease of designating mapping resolution with consistent inter-indent spacing, more consistent indentation strain rates, and more consistent sampled volumes for correlation to analytical electron microscopy techniques such as EDS or EBSD. In this application note, we explore the differences in nanoindentation mapping results that arise from the use of displacement or load control during testing on a two-phase Al—Cu eutectic sample [1]. The variation in indentation sizes within the two phases in this material between the control modes is discussed in relation to the achieved loads and contact areas and stiffnesses in each phase. The morphology variation in the resulting maps and their property distributions are examined, and the general implications of displacement and load control on indentation mapping results are deconvoluted.

Figure 1. Comparison of indentation maps of hardness and reduced modulus performed in displacement and load control on Al—Cu eutectic sample plotted on matching scales. Load control map taken from subset of literature data [1] for matching scales for visual comparison.

Experimental Considerations

To ensure a close comparison, both sets of results are acquired on the same Al—Cu eutectic sample in the same state of surface preparation, and both sets of data were performed with the same target indentation depth of 100 nm and an inter-indentation spacing of 1 µm. The Al-Cu eutectic alloy with a composition of Al-33 wt% Cu was prepared from Al and Cu pure metals with purity of 99.95% (Zhongnuo New Materials Ltd.). The pure metals were melted using vacuum arc melting and then solidified in water-cooled copper crucible. The sample was then sectioned and metallographically prepared using standard methods. Conventional CSM nanoindentation and load-controlled indentation mapping results were reported in the literature [1].

Mechanical microscopy was performed using an FT-I04 Femto-Indenter (FemtoTools AG, Switzerland) with a diamond Berkovich indenter. Each indentation was conducted in ~1 s with an oscillation amplitude of 1 nm at a frequency of 140 Hz with the oscillation amplitude linearly increased with penetration depth. Each indentation was performed to a specified depth of 95 nm, so that a spacing of 1 µm between indentations could be used while still ensuring an indentation depth/spacing ratio of 10 was maintained [3]. This avoids any significant interaction between neighboring indents. Hardness and reduced modulus were measured as a function of depth for each location, and representative values for each were taken by averaging values from depths > 60 nm.

Results and Discussion

Map Comparison

 For an initial comparison between the two control modes, we first examine the appearance of the resulting indentation property maps from both datasets - Figure 2. The 100×100 indentation size of each map is a selected subset of each map, which were both acquired with larger but not matching dimensions, chosen to allow a direct visual comparison between them.

Figure 2. Comparison of indentation mapping results measured using load control [1] and displacement control on an Al-Cu eutectic sample plotted as 2D histograms of achieved loads vs. contact depths and hardness vs. reduced modulus.

At first glance, maps from both control modes appear quite similar in terms of morphology. Both display the fine microstructural features of the eutectic microstructure. As both maps are performed with the same inter-indent spacing, this may be somewhat expected. However, on closer inspection of the single-phase regions, one can note that the maps acquired in displacement control display much higher consistency/uniformity in values inside the single-phase areas of both Aluminum and Al₂Cu. This is particularly notable in the harder Al₂Cu phase and in the reduced modulus maps. Differences between the values measured for each phase are discussed more in depth in Section 3.3.

Contact Depth and Load Distributions

Before looking into the differences in the measured material properties, it’s instructive to first examine the achieved contact depths and loads from each control mode - Figure 2.

In load control, a maximum load target provides a precisely determined maximum load very near to the 0.4 mN target in both phases. However, due to the significant difference in hardness between the Al₂Cu and Aluminum phases, the scatter in mean contact depths in the two phases is > 2×, with a total variation from 30 to 160 nm, which ensures a proportionally larger variation in contact areas and contact stiffnesses. It is expected that the variation in indentation depths in load control is proportional to the square root of the hardness variation between the two phases. As stated previously, the max. load of 0.4 mN was selected to ensure that all indentations have a contact depth of less than 100 nm to satisfy the 10 d/h criterion for the 1 µm spacing. However, Figure 2 reveals that this was partly unsuccessful, suggesting some indentations in the softer Al phase likely interacted. This implies that a lower load should have been selected, which would’ve resulted in even shallower indentations in the harder phase.

In displacement control, using a maximum depth target of 95 nm, all indentations are seen to have contact depths (after elastic unloading) of less than 100 nm, with the contact depth decreasing with the hardness/modulus ratio of the phases due to elastic recovery. This ensures that no interaction occurred between neighboring indentations in displacement control if the 10 d/h criterion is appropriate.

Neither displacement control nor load control modes produce indentations with uniform contact depths due to elasticity, but the indentation depth/size variation is significantly reduced using displacement control. This has ramifications when comparing results between the phases due to indentation size effects. The magnitude of indentation size effects varies among various materials significantly. Well-annealed, soft metallic single crystals can display relatively large size effects and work-hardened, hard, or nanocrystalline materials display relatively small effects [24]. To ensure that an absolute comparison could be made between the properties of dissimilar phases at a selected indentation size, indentation mapping would ideally be performed using a constant indentation strain rate using a continuous stiffness measurement technique, so that data can be selected from matching depths for comparison under the same experimental conditions.

Perhaps even more instructive than the variation in contact depths is the disparity between the attained loads. This is precisely set in load control at 0.4 mN. However, attained load values for the max. depth target shows a vast spread, with the harder Al₂Cu phase producing loads over 4× greater than the selected maximum load used for load-controlled mapping. This is consistent with expectations, as a material 4× harder would require 4× greater loads for the same indentation area. Thus, for a nanoindentation system with identical measurement sensitivity using either displacement or load control, it is clear that indentation mapping using displacement control would provide much greater and more consistent measurements from a microstructure with dissimilar phases. This is due to greater contact depths, loads, and contact stiffnesses achieved in the harder phases while still satisfying the spacing criterion to prevent overlapping indentations in the softer phase.

Hardness and Modulus Distributions

 Hardness and reduced modulus value distributions resulting from the two control modes support this assertion - Figure 2. Reduced modulus values were recalculated from [1] to remove the assumed Poisson’s ratio from the previously plotted elastic modulus values. These distributions are plotted from the full datasets for each control mode with the total number of indentations, N, being 40,000 in load control and 12,800 in displacement control. Despite the larger statistical sample for load control, significantly more precise property distributions are observed in the dataset acquired with displacement control.

Further, using displacement control, the property distributions are more accurately centered at values consistent with conventional CSM indentation values measured at contact depths of 100 nm [1]: Aluminum phase – H = 1.87 and E_r = 83.8 GPa, and Al₂Cu phase – H = 6.81 and E_r = 122.5 GPa. While these conventional CSM values were acquired from a small number of manually-selected single-phase regions, the values are also consistent with those acquired on an Al–Cu diffusion couple investigated by the nanoindentation [4]. The variation of the mapping results under load control from these values is discussed at some length elsewhere [1]. The results from displacement control mapping in Figure 2 show distributions centered quite closely to these reference values. Reduced modulus distributions for both phases show peaks (brightest 2D histogram bins) in excellent agreement with these values. Hardness values for the Aluminum phase show a distribution which includes these values and also extends somewhat lower, while hardness values in the Al₂Cu phase start at similar values to the reference and extend to higher values. The reason for this is possibly due to strain rate or orientation effects.

Influence of Surface Contamination

The influence of contamination on mapping with load or displacement targets is also worthy of note. Indentation mapping in load control is less prone to measurement errors from the presence of soft debris over the tested surface. The maximum depth achieved during mapping in load control depends less on detecting the initial contact displacement, so surface contamination is less influential.

During operation in displacement control, an initial contact detection threshold must be set in terms of measured load/stiffness/et cetera to determine the contact displacement, such that a target maximum displacement can be precisely achieved. This contact threshold can be set to a higher level to increase tolerance to contamination, but this decreases the precision to which the depth target can be achieved in either load or displacement-targeted operation. When a higher threshold detection load is used, it is strongly suggested that matching displacement rates be used for the approach and indentation phases of testing.

Contamination may accumulate on the indenter during mapping, typically at a depth similar to the median max. depth. This contamination can act to increase the local contact area between the indenter and sample and the apparent contact stiffness. Thus, it is important to maintain indenter and sample cleanliness, particularly for indentation mapping at lower penetration depths.

Conclusion

In this application note, nanoindentation mapping results on an Al—Cu eutectic sample were compared to determine the influence of load and displacement control modes on the quality of the results. While the morphology of the resulting maps was comparable, due to the matching resolution with 1 µm spacing between the indentations in both grids, there were significant differences notable in the values and statistical distributions produced by each control mode.

In terms of achieved contact depths and loads, the two different control modes produce very contrasting results. In load control, the load is very precisely determined for both phases, but this results in a variation in indentation depths proportional to the square root of the hardness variation between the two phases. In displacement control, the load, rather than the indentation depth, is expected to scale proportionally to the hardness of each phase, as the contact area is kept as constant as possible. This was observed, as the harder Al₂Cu phase produced loads over 4× greater than the selected maximum load used for load-controlled mapping. A smaller variation in contact depth is still achieved in displacement control due to elastic recovery.

Thus, for any nanoindentation system with identical measurement sensitivity (in terms of force or displacement resolution) using either displacement or load control, it is clear that indentation mapping using displacement control is expected to provide measurements with superior accuracy and precision. This is due to greater contact depths, loads, and contact stiffnesses achieved in the harder regions while still satisfying the spacing criterion to prevent overlapping indentations in the softer regions. This is also supported by the present mapping results in displacement control displaying close agreement with reference values, particularly in terms of reduced modulus. While the performance of indentation mapping using load control does provide a nice visual reference for the hardness of different regions within the indentation grid by way of indentations in harder phases displaying smaller residual indentations, we recommend always performing indentation mapping in displacement control to obtain superior quality results.

References

1. H. Besharatloo, J.M. Wheeler, Journal of Materials Research, 36 (2021) 2198-2212.
2. L.E. Samuels, T.O. Mulhearn, J. Mech. Phys. Solids, 5 (1957) 125-134.
3. P.S. Phani, W. Oliver, Materials & Design, 164 (2019) 107563.
4. Y. Xiao, H. Besharatloo, B. Gan, X. Maeder, R. Spolenak, J.M. Wheeler, Journal of Alloys and Compounds, 822 (2020) 153536.