Theory of the oscillatory instability of a rigid tip scraped over a polymer surface

The patterns produced by dragging an atomic force microscope (AFM) tip over a polymer surface are studied using a mesoscopic model introduced by Gnecco and co-workers [E. Gnecco et al., New J. Phys. 17, 032001 (2015)1367-263010.1088/1367-2630/17/3/032001]. We show that the problem can be reduced to solving a closed integrodifferential equation for a single degree of freedom, the position of the AFM tip. We find the steady-state solution to this equation and then carry out a linear stability analysis of it. The steady state is only stable if the dimensionless indentation rate α is less than a critical value α_{c} which depends on the dimensionless velocity of the rigid support r. Conversely, for α>α_{c}, periodic stick-slip motion sets in after a transient. Simulations show that the amplitude of these oscillations is proportional to (α-α_{c})^{1/2} for α just above α_{c}. Our analysis also yields a closed equation that can be solved for the critical value α_{c}=α_{c}(r). If the steady-state motion is perturbed, as long as the deviation from the steady state is small, the deviation of the tip's position from the steady state can be written as a linear superposition of terms of the form exp(λ_{k}t), where the complex constants λ_{k} are solutions to an integral equation. Finally, we demonstrate that the results obtained for the two-dimensional model of Gnecco et al. carry over in a straightforward way to the generalization of the model to three dimensions.