# Nanoindentation of Historic and Artists' Paints
## 4 Characterization of Artists' Paints
### Free Film Paint Samples
#### 4.1.3 Contact between Indenter and Sample  
Surface roughness is not the only source of error in determining the contact depth (\(h_{\text{c}}\)), and, consequently, the projected contact area (\(A_{\text{p}}\)) and the equivalent contact radius (\(a=\sqrt{A_{\text{p}}/\pi}\)). The Oliver-Pharr method calculates the contact depth with a pyramidal or conical indenter as \(h_{\text{c}}=h_{\text{max}}-0.75P_{\text{max}}/S\), assuming that the deformation of the sample occurs solely in elastic manner [18]. However, many materials deform both elastically and plastically upon indentation by a diamond Berkovich indenter. Fortunately, the contact equation works reasonably well for an elasto-plastic solid that work-hardens appreciably during the indentation test [31]. For a non-work-hardening solid, however, the equation does not work well as the material 'piles up' around the indenter. The surface profile of a material with 'pile-up', shown schematically in Figure 5b for a conical indenter geometry, is a consequence of the plastic flow of the material outside the contact perimeter. Since the contact equation assumes the absence of 'pile-up' (Figure 5a), it is bound to underestimate the contact depth in the presence of 'pile-up' [18; 31]. As a result, the nanoindentation-derived modulus (\(\propto 1/a\)) and hardness (\(\propto 1/A_{\text{p}}\)) of the specimen are overestimated. The extent to which a material 'piles up' can be pronounced when testing a thin film constrained by a much harder and stiffer substrate [43]. The occurrence of 'pile-up' can be confirmed by microscopic (optical or AFM) observation of the indentation imprint. If 'pile-up' has occurred during the test with a pyramidal indenter, the sides of the residual impression appear bowed-out, whereas excessive'sinking-in' results in the sides of the impression bowing-in, as shown in Figure 6a. The contact equation works well for the residual imprints on the left and in the middle of Figure 6a but underestimates the contact depth for the imprint on the right. Theoretically, the Oliver-Pharr method can also overestimate the contact depth in the presence of excessive'sink-in' [43]. Such a situation may occur while testing a hard film on a soft substrate, a rather unusual situation for paint samples. Paints, especially those which are soft, can be notoriously prone to 'pile-up' issues. Since the properties of paints depend on temperature and relative humidity, the extent of 'pile-up' cannot be discussed without considering the environmental conditions during the test. The residual imprints of Winsor & Newton ultramarine oil paint [11] with a Berkovich indenter, presented in Figure 6b, show the changing nature of the probe-material contact with increasing relative humidity. 'Sink-in' observed in the tests performed at 35%RH is no longer present at 60%RH. At 70%RH there is a clear manifestation of 'pile-up', whereas at 83%RH the indenter leaves a shapeless imprint, indicative of the paint's liquid-like properties in this environment. Indentation testing of the ultramarine paint at high humidity, when analyzed using the Oliver-Pharr method, results in systematic  
Figure 4: (**a**) AFM scan of a laboratory prepared TiO\({}_{2}\) acrylic paint, and (**b**) two-dimensional surface profile of the paint along the thick segment of the red line in (**a**) contacted by an indenter with a half-included angle of 70.3\({}^{\circ}\), showing uncertainty in defining the contact depth on a rough surface. Note that the aspect ratio of the indenter (indicated by the blue triangles) is correctly scaled with the surface profile. overestimation of the paint modulus. The extent of this overestimation is difficult to assess without precise recording of the additional contact area created by 'pile-up'. This, however, is an extremely difficult, if not impossible, task for environmentally-softened pains as the indentation imprint may change with time, both with respect to shape and size, due to the pronounced time dependence of their mechanical properties.