Nanoindentation, also called instrumented indentation testing, is a variety of indentation hardness tests applied to small volumes. Indentation is perhaps the most commonly applied means of testing the mechanical properties of materials. The nanoindentation technique was developed in the mid-1970s to measure the hardness of small volumes of material.
Background In a traditional indentation test (macro or micro indentation), a hard tip whose mechanical properties are known (frequently made of a very hard material like diamond) is pressed into a sample whose properties are unknown. The load placed on the indenter tip is increased as the tip penetrates further into the specimen and soon reaches a user-defined value. At this point, the load may be held constant for a period or removed. The area of the residual indentation in the sample is measured and the hardness, , is defined as the maximum load, , divided by the residual indentation area, :
For most techniques, the projected area may be measured directly using light microscopy. As can be seen from this equation, a given load will make a smaller indent in a "hard" material than a "soft" one.
This technique is limited due to large and varied tip shapes, with indenter rigs which do not have very good spatial resolution (the location of the area to be indented is very hard to specify accurately). Comparison across experiments, typically done in different laboratories, is difficult and often meaningless. Nanoindentation improves on these macro- and micro-indentation tests by indenting on the nanoscale with a very precise tip shape, high spatial resolutions to place the indents, and by providing real-time load-displacement (into the surface) data while the indentation is in progress.
In nanoindentation small loads and tip sizes are used, so the indentation area may only be a few square micrometres or even nanometres. This presents problems in determining the hardness, as the contact area is not easily found. Atomic force microscopy or scanning electron microscopy techniques may be utilized to image the indentation, but can be quite cumbersome. Instead, an indenter with a geometry known to high precision (usually a Berkovich tip, which has a three-sided pyramid geometry) is employed. During the course of the instrumented indentation process, a record of the depth of penetration is made, and then the area of the indent is determined using the known geometry of the indentation tip. While indenting, various parameters such as load and depth of penetration can be measured. A record of these values can be plotted on a graph to create a load-displacement curve (such as the one shown in Figure 1). These curves can be used to extract mechanical properties of the material.
Young's modulus The slope of the curve, , upon unloading is indicative of the stiffness of the contact. This value generally includes a contribution from both the material being tested and the response of the test device itself. The stiffness of the contact can be used to calculate the reduced Young's modulus :
where is the projected area of the indentation at the contact depth , and is a geometrical constant on the order of unity. is often approximated by a fitting polynomial as shown below for a Berkovich tip:
where for a Berkovich tip is 24.5 while for a cube corner (90°) tip is 2.598. The reduced modulus is related to Young's modulus of the test specimen through the following relationship from contact mechanics:
Here, the subscript indicates a property of the indenter material and is Poisson's ratio. For a diamond indenter tip, is 1140 GPa and is 0.07. Poisson's ratio of the specimen, , generally varies between 0 and 0.5 for most materials (though it can be negative) and is typically around 0.3.
There are two different types of hardness that can be obtained from a nano indenter: one is as in traditional macroindentation tests where one attains a single hardness value per experiment; the other is based on the hardness as the material is being indented resulting in hardness as a function of depth.
Hardness The hardness is given by the equation above, relating the maximum load to the indentation area. The area can be measured after the indentation by in-situ atomic force microscopy, or by 'after-the event' optical (or electron) microscopy. An example indentation image, from which the area may be determined, is shown at right.
Some nanoindenters use an area function based on the geometry of the tip, compensating for elastic load during the test. Use of this area function provides a method of gaining real-time nanohardness values from a load-displacement graph. However, there is some controversy over the use of area functions to estimate the residual areas versus direct measurement. An area function typically describes the projected area of an indent as a 2nd-order polynomial function of the indenter depth . When too many coefficients are used, the function will begin to fit to the noise in the data, and inflection points will develop. If the curve can fit well with only two coefficients, this is the best. However, if many data points are used, sometimes all 6 coefficients will need to be used to get a good area function. Typically, 3 or 4 coefficients works well. Service Document Probe Calibration; CSV-T-003 v3.0; Exclusive application of an area function in the absence of adequate knowledge of material response can lead to misinterpretation of resulting data. Cross-checking of areas microscopically is to be encouraged.
Strain-rate sensitivity The strain-rate sensitivity of the flow stress is defined as
where is the flow stress and is the strain rate produced under the indenter. For nanoindentation experiments which include a holding period at constant load (i.e. the flat, top area of the load-displacement curve), can be determined from
The subscripts indicate these values are to be determined from the plastic components only.
Activation volume Interpreted loosely as the volume swept out by dislocations during thermal activation, the activation volume is
where is the temperature and kB is the Boltzmann constant. From the definition of , it is easy to see that .
Hardware Sensors The construction of a depth-sensing indentation system is made possible by the inclusion of very sensitive displacement and load sensing systems. Load transducers must be capable of measuring forces in the micronewton range and displacement sensors are very frequently capable of sub-nanometre resolution. Environmental isolation is crucial to the operation of the instrument. Vibrations transmitted to the device, fluctuations in atmospheric temperature and pressure, and thermal fluctuations of the components during the course of an experiment can cause significant errors.
Continuous stiffness measurement (CSM)
Dynamic nanoindentation or continuous stiffness measurement (CSM, also offered commercially as CMX, dynamics...), introduced in 1989, is a significant improvement over the quasi-static mode described above. It consists into overlapping a very small, fast (> 40 Hz) oscillation onto the main loading signal and evaluate the magnitude of the resulting partial unloadings by a lock-in amplifier, so as to quasi-continuously determine the contact stiffness. This allows for the continuous evaluation of the hardness and Young's modulus of the material over the depth of the indentation, which is of great advantage with coatings and graded materials. The CSM method is also pivotal for the experimental determination of the local creep and strain-rate dependent mechanical properties of materials, as well as the local damping of visco-elastic materials. The harmonic amplitude of the oscillations is usually chosen around 2 nm (RMS), which is a trade-off value avoiding an underestimation of the stiffness due to the "dynamic unloading error" or the "plasticity error" during measurements on materials with unusually high elastic-to-plastic ratio (E/H > 150), such as soft metals.
Atomic Force Microscopy The ability to conduct nanoindentation studies with nanometre depth, and sub-nanonewton force resolution is also possible using a standard AFM setup. The AFM allows for nanomechanical studies to be conducted alongside topographic analyses, without the use of dedicated instruments. Load-displacement curves can be collected similarly for a variety of materials – provided that they are softer than the AFM tip – and mechanical properties can be directly calculated from these curves. Conversely, some commercial nanoindentation systems offer the possibility to use a piezo-driven stage to image the topography of residual indents with the nanoindenter tip.
Optical Interferometry Utilizing optical fiber Fabry-Perôt interferometry, nanoindentation studies can be performed with unparalleled precision, achieving micro-mechanical characterization of soft biomaterials. Optical interferometry, allows for nanomechanical studies of biomaterials alongside topographic analyses without the need for dedicated instruments. This technology is particularly adept at micro-mechanical analysis of soft and living materials. Load-displacement curves can be gathered for a diverse range of materials, and their mechanical properties can be directly inferred from these curves. Additionally, some advanced systems offer the capability to integrate optical imaging with micro-mechanical characterization, enabling a comprehensive understanding of the relationship between structure and stiffness in biomaterials.
Software Experimental software The indentation curves have often at least thousands of data points. The hardness and elastic modulus can quickly be calculated by using a programming language or a spreadsheet. Instrumented indentation testing machines come with the software specifically designed to analyze the indentation data from their own machine. The Indentation Grapher (Dureza) software is able to import text data from several commercial machines or custom made equipment.David Shuman; Indent Grapher (Dureza) Software Spreadsheet programs such as MS-Excel or OpenOffice Calculate do not have the ability to fit to the non-linear power law equation from indentation data. A linear fit can be done by offset displacement so that the data passes through the origin. Then select the power law equation from the graphing options.
The Martens hardness, , is a simple software for any programmer having minimal background to develop. The software starts by searching for the maximum displacement, , point and maximum load, .
The displacement is used to calculate the contact surface area, , based on the indenter geometry. For a perfect Berkovich indenter the relationship is .
The indentation hardness, is defined slightly different.
Here, the hardness is related to the projected contact area .
As the indent size decreases the error caused by tip rounding increases. The tip wear can be accounted for within the software by using a simple polynomial function. As the indenter tip wears the value will increase. The user enters the values for and based on direct measurements such as SEM or AFM images of the indenter tip or indirectly by using a material of known elastic modulus or an atomic force microscope (AFM) image of an indentation.