Interpreting Hardness Data

By measuring the size of an indentation made by a measured force, it is possible to estimate the hardness of the sample surface. This is analogous to industrial hardness tests (e.g., Rockwell and Brinell tests) however, at the nanometric level precise one-to-one comparisons are elusive.

Force Calculations for Indentation

To calculate the force applied to the sample during indentation it is necessary to first measure the cantilever sensitivity. The cantilever sensitivity, called the Deflection sens. or TM Deflection Sens. in the software, is the cantilever deflection signal versus the voltage applied to the Z piezo determined from the slope of the force plot during indentation. The cantilever sensitivity is a conversion factor from cantilever deflection in volts to cantilever deflection in nm. Its units are nm/volt. Once the sensitivity is known, the force (F) of indentation is determined from Hooke's Law:

where k is the spring constantAmount of force required to bend a cantilever a specific amount (typical units are N/m). in N/m and x is the cantilever deflection (for this particular case). The cantilever deflection is determined from the cantilever sensitivity and the maximum deflection used for the indentation (Trigger threshold).

Measuring Cantilever Sensitivity

It is more accurate to measure the cantilever sensitivity on a very hard sample, since there is little surface penetration. Since the cantilever sensitivity is a measurement of the change in cantilever deflection per change in the Z position, it is important to have negligible sample penetration. If a change in Z position causes not only cantilever deflection but also sample penetration, the sensitivity measurement is inaccurate. For a soft sample, more Z motion is required to obtain the same deflection as on a hard sample. Thus the sensitivity is greater when measured on a harder sample.

The ideal sensitivity would be obtained on an infinitely hard sample. The ideal sensitivity is approached as the sample penetration approaches zero or as the hardness becomes infinite. It is suggested that a hard sample, such as the sapphire sample provided with the nanoindentation package, is used to measure the cantilever sensitivity before or after indenting the desired sample. The more accurate you need your calculation of the indentation force to be, the more important this becomes.

The procedures for Force Calibration vary according to the mode selected. For detailed force calibration information, including procedures, see:

• Contact Mode Force Calibration
• TappingMode Force Calibration

Notes on Comparing Hardness Data

Nanoindentation is best suited for making relative, rather than absolute, hardness measurements. When making hardness comparisons, it is imperative that conditions be replicated as best as possible, preferably using the same diamond tip without repositioning the tip or changing the position of the laser spot on the cantilever during the tests. If it is necessary to compare many samples or many locations within a sample, or if the tip is replaced, note the following:

• All things being equal (cantilever spring constant, nanoindentation parameters, etc.) a sharp diamond will make a deeper indentation than a blunt diamond. This means that the same amount of force applied to two different diamonds will not make the same size indentation. Therefore, indentation size alone should not be used to calculate hardness data between two or more tips. Although the same sample may be used as a relative standard between two or more tips, bear in mind that the diamonds employed may differ significantly.
• When changing diamond tips, or when reinstalling the same tip, the cantilever sensitivity must be checked and reset each time.
• When comparing hardness results between two or more tips, mathematical modeling provides the best approximation of force and, therefore, hardness. Applying the F = –kx formula described above, one may use the following example as a template:

That is, by setting the Trigger threshold of tip “B” to 0.8 V, the same amount of force should be delivered to the surface as when using tip “A.” By observing dents creating by the two tips on the same sample surface, it is possible to obtain a subjective comparison. For this reason, it is useful to keep a “touchstone” nanoindentation standard. Recall that indentation sizes vary depending upon the sharpness of the diamond, even for tips mounted on cantilevers having the same spring constant. If the observed dents are grossly different, recheck the calculations—the Trigger threshold may be set incorrectly.

• It may prove difficult to apply the same force using a new tip with a much lower spring constant. For example, if switching from a “harder” tip (or stiffer cantilever) to a much “softer” tip (or less stiff cantilever), it may be impossible to push with sufficient force to produce a comparable indentation. In this case, it would be better to either use a harder tip, or rerun samples using the softer tip to obtain consistent results across all samples.

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