Contact angle dependence of solid probe–liquid drop forces in AFM measurements

In order to obtain the force between an atomic force microscopy probe particle and a liquid drop, we require the small change δR in the mean radius of curvature of the drop, which is proportional to the small change in liquid pressure inside the drop. These quantities depend on the total force F, the contact angle θ made by the unperturbed drop at the substrate, and the nature of the boundary conditions imposed at the contact line. Chan et al. [J. Colloid Interface Sci. 236 (2001) 141] developed a semi-analytical method for calculating the force in the fixed contact line case, while more recently Attard and Miklavcic [Langmuir 17 (2001) 8217] considered the fixed contact angle condition. Due to errors in perturbation theory calculations, the results presented by these authors for δR are incorrect, and their results [J. Colloid Interface Sci. 236 (2001) 141; Langmuir 17 (2001) 8217; J. Colloid Interface Sci. 247 (2002) 255] for the force are thus also in error. In contrast with the results of the above authors, we find that δR is singular at θ=0, and perturbation theory breaks down at small θ. Consequently we find that, for angles of practical interest, δR and thus the change in fluid pressure inside the drop can be orders of magnitude larger than previous calculations suggest. Modifying the method from [J. Colloid Interface Sci. 236 (2001) 141] accordingly, we present indicative numerical force F(X) curves (where X is the central probe-stage separation) for both boundary conditions. The general perturbation expansions are invalid near θ=π/2. We therefore extend the results into this experimentally significant region with a separate analysis at θ=π/2. Additionally, a nonperturbative matched expansion is developed to extend the results to small θ and to verify all analytical results.