Single molecule manipulation using techniques such as atomic force microscopy (AFM), optical and magnetic tweezers enable detailed study of geometry and forces in long polymers [1].When a probe with a scale invariant shape is close to a long polymer (separation h much smaller than polymer size), the force between the polymer and the probe is F = A k_bT/h. Kantor et al. showed that the universal amplitude A can be related to correlation exponents of long polymers [2].

We find the force amplitude for phantom polymers in several scale invariant geometries (wedges, cones, tilted cones, cones with various cross sections), both analytically by solving the equation for the correlation function, and from numerical simulations. In the latter approach we utilize the simple connection between the average polymer end-to-end distance and the force on the surface. We derive the local pressure exerted by a long phantom polymer on the surface of a wedge or a cone. When the polymer is held near the tip of the object, the pressure on the tip diverges. This is similar to the divergence of electric fields near the tip of a charged conductor.

[1] T.-H. Nguyen, S.-M. Lee, K. Na, S. Yang, J. Kim and E.-S. Yoon, Nanotechnology 21, 075101 (2010).

[2] M. F. Maghrebi, Y. Kantor and M. Kardar, Europhys. Lett. 96, 66002 (2011).