Building and Fire Research Laboratory

Nanoscience of Polymers Topics:
	Atomic Force Microscopy
	Nanoindentation of Polymers
	Nanoparticle Dispersion
	Polymer Interphases Consortium

Nanoindentation of Polymers

The analysis of indentation load-penetration curves produced by depth-sensing indentation systems is often based on work by Oliver and Pharr. Their analysis was in turn based upon relationships developed by Sneddon for the penetration of a flat elastic half space by different probes with particular axisymmetric shapes (e.g., a flat-ended cylindrical punch, a paraboloid of revolution, and a cone). In general, the relationships between penetration depth, h, and load, P, for such indenter geometries can be represented in a power law expression: P = a(h - h_f)^m, where a contains geometric constants, the sample elastic modulus, the sample Poisson's ratio, the indenter elastic modulus, and the indenter Poisson's ratio, h_f is the final unloading depth, and m is a power law exponent that is related to the geometry of the indenter; for a flat-ended cylindrical punch, m = 1, for a paraboloid of revolution, m = 1.5, and for a cone, m = 2.

In applying this equation to the calculation of modulus, Oliver and Pharr made two significant realizations. First, the slope of the unloading curve changes constantly due to a constantly changing contact area. In prior research, the high load portion of the unloading curve was approximated as linear, which incorrectly assumes that the contact area remains constant for the initial unloading of the material. This practice created a dependence of calculated modulus values on the number of points used in the linear fit. Second, if the unloading curve can be fit by a power law expression, then a derivative, dP/dh, applied at the maximum loading point (h_max, P_max) should yield information about the state of contact at that point. This derivative was termed the contact stiffness, S, which is a function of the projected area of tip-sample contact, A, and the reduced modulus, E_r, which contains the elastic modulus (E) and Poisson's ratio (n) for both the sample and the indenter material and accounts for deformation of both the indenter and the sample.

In the figure above, an indentation load-displacement curve is illustrated along with several important parameters used in the Oliver and Pharr analysis. The stiffness, S*, is the slope of the tangent line to the unloading curve at the maximum loading point (h_max, P_max). When the displacement, h, is the total measured displacement of the system, S* is the total system stiffness. After successful calibration and removal of the load-frame compliance, the displacement of the load frame is removed so that h represents only the displacement of the tip into the sample. In this case, S* = S and the tangent line represents an unloading path for which the contact area does not change. Thus, the contact area, A, calculated using S should be the actual contact area at maximum load. Also, extrapolating this line down to P = 0 yields an intercept value for depth, h_i, which should be related to the contact depth, h_c, associated with the maximum loading point. However, h_c is related to the deformation behavior of the material and the shape of the indenter, as illustrated in the figure. In fact, h_c = h_max - h_s, where h_s is defined as the elastic displacement of the surface at the contact perimeter and can be calculated for specific geometries using displacement equations from Sneddon's analyses. For each of three specific tip shapes (flat-ended punch, paraboloid of revolution, and cone), h_s = eP_max/S where e is a function of the particular tip geometry, as summarized in the table below.

Theoretical values of m and e for three axisymmetric tip shapes.

Tip Geometry	m	e
Flat-ended cylindrical punch	1	1
Paraboloid of revolution	1.5	0.75
Cone	2	2(p-2)/p

The nanoindentation procedures include calibration of the load-frame compliance, C_lf, and the tip shape area function, A(h_c). A number of possible methods exist for determining C_lf and A(h_c) using a reference sample that is homogeneous and isotropic and for which both E and n are known. Typically, a series of indentation measurements are made on the reference sample. Oliver and Pharr suggested using an iterative technique to calibrate both the load-frame compliance and the tip shape with one set of data from a single reference sample. While this method has the advantage of not requiring an independent measurement of the area of each indent, its use has been limited, perhaps because it is mathematically intensive. For the load-frame compliance calibration, relatively large indentation loads and depths are applied to a reference material that exhibits significant plastic deformation (e.g. aluminum) so that the contact stiffness is large and thus total compliance is dominated by the load-frame compliance. For the tip shape calibration, the series of indents applied to a reference material typically covers a larger range of maximum load and maximum penetration depth. The objective of tip shape calibration is to measure the cross-sectional area of the indenter tip as a function of distance from the apex. At a given load, P, the contact area, A, is the cross-sectional area of the indenter tip at a distance, h_c (the contact depth), from the tip apex. From measurements of h_max, P_max, and S, A and h_c are calculated for each indentation. A tip shape function, A(h_c), is determined, given a sufficient number of measurements over a range of h_c values, by fitting the A vs. h_c data, typically using a multiterm polynomial fit.

Once the load frame compliance and tip shape calibrations have been performed, measurements of elastic modulus for samples of interest can be made from indentation data. The unloading curves are again fit to a power law function, and the fitting parameters are used to calculate S*, which is equal to S assuming a correctly determined value of C_lf. S is then used to calculate h_c, and h_c is used to calculate A from the tip shape area function. Finally, S and A are used to calculate E.

DSI systems, also referred to as nanoindenters in some cases, are much more robust tools for performing indentation measurements compared to the AFM. However, because these systems have traditionally been used to indent hard inorganic engineering materials (e.g., metals, ceramics, and semiconductors), applications to polymeric materials have been limited. In particular, the lowest detectable load level is generally around 1 mN. However, significant deformation of polymers can occur at this load level. In fact, typical applied forces of DSI systems range from 100 mN up to 10 mN or higher. These types of forces produce large penetration depths (> 100 nm) and residual impressions (lateral dimensions of 1 mm or larger), depending on the polymer, the force level, and the tip shape. For the purposes of our research, property measurements of organic engineering materials (i.e., polymeric and biological materials) with nanoscale spatial resolution are desired. Also, elasticity theory is used to calculate values of elastic modulus from indentation data. However, because polymeric and biological materials exhibit time-dependent rheology, the use of elasticity theory to measure the modulus for these materials will likely fail. Finally, material-dependent calibration procedures are used in DSI to measure load-frame compliance and tip shape. In both cases, large experimental uncertainties can occur that propagate through calculations of material properties. For example, in a study performed using a DSI system and a diamond indentation tip with Berkovich geometry, blind reconstruction measurements of the tip area function did not agree with indentation tip shape calibration using a fused silica reference sample. While the blind reconstruction measurements contain some uncertainty, this discrepancy is likely due to uncertainties associated with the indentation tip shape calibration procedure, in particular the load-frame compliance as shown below.

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	AFM Indentation

Building and Fire Research Laboratory

Nanoindentation of Polymers

Depth-Sensing Indentation (DSI)

DSI Basics

Application to Polymers

References:

Recent Results


Left -- An indentation load-displacement curve in which several important parameters used in the Oliver and Pharr analysis are illustrated.; Right -- Illustration of the indentation geometry at maximum load for an ideal conical indenter.


Left -- scanning electron micrography (SEM) of a spike characterizer sample used for blind reconstruction; Right -- AFM topographic image of the spike characterizer sample scanned with the Berkovich tip (total height scale in the image is 800 nm from black to white).

Left -- 3D representation of blind reconstruction data of a Berkovich indenter tip generated using AFM images taken with the Berkovich tip on several tip characterizer samples, such as the spike characterizer sample in the previous images; Right -- comparison of area functions for blind reconstruction and indentation tip shape calibration for the DSI Berkovich tip, where the indentation tip shape calibration was performed using a fused silica reference sample; black dashed curves represent +/- 2s for the tip 2 data, where s is an estimated standard deviation based on uncertainty in the load-frame compliance calibration.