The free energy of long polymers is frequently dominated by entropy with the interaction energy playing a minor role. In the absence of an energy scale, the corresponding forces are then governed by the thermal energy scale k_BT and by the length scales associated with the experimental set-up. Recent advances in single molecule manipulation techniques have brought the accuracy of position and force determination into the range where forces exerted on a surface by a single weakly deformed molecule can be measured. At such deformations shapes of the surfaces to which the polymer is attached play an important role. If these surfaces lack definite length scale (such as plane, cone, line, etc.) forces between the polymer and the surface can be frequently evaluate exactly. E.g., if a cone with a polymer attached to its sharp tip approaches a plate, then for cone-plate separation h<<R₀ the polymer-mediated force between them [1] is given by F=Ak_BT/h. The coefficient A can be related to geometry-dependent correlation exponents of long polymers. We computed A for phantom polymers, and for self-avoiding polymers by ε-expansion, as well as by numerical simulations in 3 dimensions. Similar results can be obtained for a large variety of surface shapes.

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