We consider a single living semi-flexible filament with persistence length lp in chemical equilibrium with a solution of free monomers at fixed monomer chemical potential μ1 and fixed temperature T. While one end of the filament is chemically active with single monomer (de)polymerization steps, the other end is grafted normally to a rigid wall to mimic a rigid network from which the filament under consideration emerges. A second rigid wall, parallel to the grafting wall, is fixed at distance L << lp from the filament seed. In supercritical conditions where monomer density ρ1 is higher than the critical density ρ1c, the filament tends to polymerize and impinges onto the second surface which, in suitable conditions (non-escaping filament regime), stops the filament growth. We first establish the grand-potential Ω(μ1,T,L) of this system treated as an ideal reactive mixture, and derive some general properties, in particular the filament size distribution and the force exerted by the living filament on the obstacle wall. We apply this formalism to the semi-flexible, living, discrete Wormlike chain model with step size d and persistence length lp, hitting a hard wall. Explicit properties require the computation of the mean force f ̄(L) exerted by the wall at L and associated potential f ̄(L) = ii −dWi(L)/dL on a filament of fixed size i. By original Monte-Carlo calculations for few filament lengths in a wide range of compression, we justify the use of the weak bending universal expressions of Gholami et al. [Phys. Rev. E 74, 041803 (2006)] over the whole non-escaping filament regime. For a filament of size i with contour length Lc = (i − 1)d, this universal form is rapidly growing from zero (non-compression state) to the buckling value f (L ,l ) = π2kBTlp over a compression range much bcp 4L2c narrower than the size d of a monomer. Employing this universal form for living filaments, we find that the average force exerted by a living filament on a wall at distance L is in practice L independent andveryclosetothevalueofthestallingforceFsH =(kBT/d)ln(ρˆ1)predictedbyHill,thisexpression being strictly valid in the rigid filament limit. The average filament force results from the product of the cumulative size fraction x = x(L,lp, ρˆ1), where the filament is in contact with the wall, times the buckling force on a filament of size Lc ≈ L, namely, FsH = x fb(L; lp). The observed L independence of FsH implies that x ∝ L−2 for given (lp, ρˆ1) and x ∝ ln ρˆ1 for given (lp, L). At fixed (L, ρˆ1), one also has x ∝ l−1 which indicates that the rigid filament limit l → ∞ is a singular limit in which an infinite pp force has zero weight. Finally, we derive the physically relevant threshold for filament escaping in thecaseofactinfilaments.
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