ABSTRACT
Az(v) = cy(f) A V p Z is the change in piezodrive position due to the change in applied voltage A V p z . Let A z n o m ( i / ) = a(v)AVpz be the nominal expansion at a different speed v\ and let pnom(vf) = A V p d / A z n o m ( ^ 0 be the nominal photodiode calibration factor at this speed, which is measured in the usual way by pressing the cantilever tip against the hard substrate. Since the correct calibration factor must be independent of speed, /3(t/) = fi(v)9 the correct expansion factor at the relevant speed can be obtained from the ratio of the measured quantities
aW) = a(v)t3nom(v')/P(v)- (1)
4.2. Water films
The van der Waals attractions fitted in Figs 4 and 5 included a parameter representing the total thickness of the water films assumed present on both surfaces. The data in Fig. 4 were obtained when the relative humidity was 31 % and that in Fig. 5 when the relative humidity was 78%. In the case of Fig. 4A, the tip jumped into contact with the silica substrate from a separation of 5.5 nm to a separation of 0.82 nm, a jump of 4.7 nm. This distance is much larger than expected for a van der Waals jump between the solids. However, water is known to condense on hydrophilic substrates [26, 27], and contact with the tip has been shown to facilitate the deposition of water and to lead to long-lived, locally thickened films that can be measured on subsequent approaches; Xu et al. [26] report induced droplets 15 nm in height and 1 /jum in radius with lifetimes of the order of minutes. If it is assumed that there is a water film on both surfaces of total thickness 3.2 nm, then quite a good description of the prejump data can be obtained using the water-air-water Hamaker constant, A = 3.7 x 10 - 2 0 J [23], and fitting a radius of curvature of the tip of 700 nm (if the Hamaker constant were doubled, on the grounds that the solids underlying the water films also contribute, then the radius of curvature would be reduced by a factor of two). The coincidence of the prejump data for the tip-silica and tip-cellulose interactions supports the assumption that the van der Waals force is due to the interaction of water films in both cases, since the van der Waals interaction of the bare solids would differ from each other.
