# Nanoindentation of Historic and Artists' Paints
## 4 Characterization of Artists' Paints
### Free Film Paint Samples
#### 4.1.2 Depth of Indentation  
The mechanical properties of many paint films can be established, despite the presence of pigment, by increasing the depth of indentation. In practice, large indentation depths may be needed to accommodate the inherent roughness of the paint surfaces. The rule of thumb dictates that the depth of indentation be at least \(\sim\)20 times the roughness depth, which amounts to an indentation depth of at least few microns for a typical paint surface. The obvious limitation of indentation depth is the maximum limit of force achievable by a nanoindenter. However, before this instrumental limit is reached, the measured mechanical response of the paint film sample may be affected by the presence of the substrate material to which it is adhered. Though a common approach to avoiding the substrate effect is to limit indentation depth to less than 10% of the film thickness, this may only help avoid large errors in establishing hardness and does not always guarantee the obtained modulus to be accurate. As clearly visible in Figure 2, the dynamic measurement of the 0.2 mm-thick acrylic paint reaches the onset of the substrate effect on \(E_{\mathrm{r}}^{\prime}\) at the contact depth of \(\sim\)5 \(\upmu\)m (\(\sim\)2.5% of the film thickness). The extent to which the substrate affects the measured response of the paint film depends on the depth of indentation relative to the film thickness, as well as the modulus mismatch ratio between the film and the substrate. Since the modulus of the paint film is unknown _a priori_, this ratio is also unknown at the stage of defining the experimental parameters. The only practical way to establish the depth range within which the mechanical response of a given paint film is free from the substrate effect is through an examination of the measured modulus as a function of depth. This can be achieved either by the multiple partial unloading indentation technique [32] or the CSM technique. In the example of the 0.2 mm-thick acrylic paint given in Figure 2, the reduced modulus remains constant from 1.5 to 5 \(\upmu\)m in contact depth, indicating that the measurement in this range is unaffected either by the surface roughness or the substrate stiffness. In the case of sharp indentation, this information can also be obtained from the measured stiffness (\(S^{\prime}\)) plotted as a function of contact depth (\(h_{\mathrm{c}}\)). For a pyramidal (or conical) indenter, the equivalent contact radius (\(a\)) increases with the contact depth; hence, the measured stiffness is a linear function of contact depth for a homogeneous material. The storage stiffness versus contact depth of the 0.2 mm-thick acrylic paint, plotted in Figure 3, indeed deviates from linearity at small (\(<\)-1.5 \(\upmu\)m) and large (\(>\)-5 \(\upmu\)m) depths due to the effects of surface roughness and substrate stiffness, respectively. The stiffness versus contact depth plot (Figure 3) also allows one to systematically and consistently establish the 'effective' contact point of the sharp indenter with a rough sample. In instrumented indentation, the initial contact point is typically defined as the instance at which the probe detects an extremely small amount of force or stiffness above a selected threshold, which is a result of the probe physically contacting the sample surface. For samples with a rough surface, however, such a procedure can result in an over- or under-estimation of the 'effective' indentation depth. To illustrate such possibilities, the surface profile of a laboratory prepared film of TiO\({}_{2}\) acrylic paint is presented in Figure 4a. The analyzed surface profile, shown in Figure 4b, was taken from a smooth area free from surface defects or irregularities. Such areas are commonly selected as preferred locations for nanoindentation testing.