Hydration Force in the Atomic Force Microscope: A Computational Study Ruoya Ho, Jian-Yang Yuan, Zhifeng Shao  Biophysical Journal  Volume 75, Issue 2, Pages 1076-1083 (August 1998) DOI: 10.1016/S0006-3495(98)77597-6 Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 1 (a) Schematic illustration of the model system used to validate the numerical method. σ is the diameter of the solvent molecule. The flat disk has a radius of 10σ and a thickness of 1σ. The total number of solvent molecules is conserved in the calculation (when the disk is moved toward the specimen, i.e., the bottom wall of the chamber). (b) The calculated hydration pressures for three different chamber sizes. □, 15σ×15σ; +, 25σ×15σ; ○, 15σ×25σ. In these calculations, a lattice size of 0.25σ was used. The results are identical, demonstrating that the effect of the side and the top of the chamber is negligible. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 2 Density profiles calculated for three different lattice sizes, 0.1σ (= σ/10, ——), 0.125σ (= σ/8, ·· ·· · ), 0.25σ (= σ/4, – – –), for a system 12σ×4σ in size (parallel plates). Clearly, even for σ/4, the accuracy of the calculation is still acceptable. The small deviation from the profiles of smaller lattices is its inability to accurately describe the spherical shape of the hypothetical solvent molecules. See text for more details. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 3 The calculated hydration force per unit area for a flat disk and a flat specimen surface at a step size of 0.25σ. The well-known oscillatory behavior is reproduced, and the quasiperiodicity of the oscillation is correlated to the size of the solvent molecule. This is identical to the results obtained by Grimson et al. (1980a), demonstrating the validity of the numerical approach developed here. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 4 The calculated hydration force for the three AFM tip shapes described in the text. (Inset) Details for the region some distance away from the surface. The solid line with squares, the dotted line with crosses, and the dashed line with circles indicate the hydration force for tip shapes 1, 2, and 3, respectively. Even for these small tips, the oscillatory behavior is still profound, and the peak positions still correspond to the size of the water molecule, although they are slightly shifted when compared with that of flat surfaces. The three curves scale precisely. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 5 A comparison of two tips with the same spherical apex (R=5σ), one with a conical shape and the other with a straight cylinder (a). The hydration force for these two tips is identical. The solid line with squares is the hydration force for tip shape 2, used in Fig. 4 (conical body), and the dashed line with pluses is that of the cylindrical tip. Therefore the hydration force is completely dominated by the apex of the tip. Thus a small asperity on a large, smooth tip surface should exhibit a hydration force as if the tip were a long needle the size of the asperity. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 6 The corresponding hydration energy calculated from the hydration force curves shown in Fig. 4. The solid line with squares, the dotted line with crosses, and the dashed line with circles indicate the hydration energies of tip shapes 1, 2, and 3, respectively. It is seen that the magnitude for shape 2 at D=1σ is very close to that determined by experiments (Cleveland et al., 1995). See text for more discussion. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 7 If an “effective contact area” is defined as the area intersected by the surface at 1σ from the specimen when the end of the tip is in touch with the specimen, the “hydration pressure” is found to be the same for all three tip shapes used in Fig. 4 (solid line with squares, dotted line with crosses, and dashed line with circles for tip shapes 1, 2, and 3, respectively). Therefore, a smaller tip will not increase the contact pressure if the hydration force dominates the interaction in imaging. It is noted that the pressure calculated here is much smaller than that of two parallel plates of the same size. Biophysical Journal 1998 75, 1076-1083DOI: (10.1016/S0006-3495(98)77597-6) Copyright © 1998 The Biophysical Society Terms and Conditions