Phantom polymer

 Calculation of γ is related to survival probability of a random walk, originating at the tip,
with appropriate (scale invariant) boundary conditions on the bounding surfaces.

 "A guide to first-passage processes," Sidney Redner, (CUP, NY, 1983)

 "Kinetics of first-passage in a cone," E. Ben-Naim, P.L. Krapivsky, J. Phys. A 43, 495007 (2010)

 The exponent η is the solution to the eigenvalue equation

Repulsive surfaces correspond to Dirichlet (absorbing) boundary condition  ψ(θ)=0 . The corresponding values of   η  are:

The resulting force amplitudes are:

Note the singular behavior as the cone angle vanishes:

Surfaces at depinning threshold correspond to Neumann (reflecting) boundary condition  ψ'(θ)=0 .

The force constant A' depends on which surface is attractive/repulsive, e.g. D cone and N plate, or vice versa, as below for d=3;

 

The interesting case of similar surface materials at the depinning transition is interesting, corresponding to   A'=0  independent of shape!