What is the distance to the wall in lattice simulations?

Lattice Monte Carlo simulations are widely used to study the effect of walls on the concentration profile in polymer solutions. The scaling theory predicts that the monomer density at a distance x from the wall, reduced by the bulk density, is proportional to (x/R_g0)^1/v at low concentrations, and the correlation length replaces R_g0 in the semidilute solution, where R_g0 is the radius of gyration and v is the Flow exponent for the chain dimension. We conducted simulations for long chains on a cubic lattice to find that a positive penetration depth γ is needed to see an agreement with the theory. The monomers perceive a theoretical wall at an off-lattice position of γ behind the presumed wall on the lattice points. We found γ ∼ 0.13 of the lattice unit at low concentrations but ∼0.36 in the semidilute solution for athermal chains. For Θ solutions, γ was 0.31-0.36 at all concentrations. We ascribe the positive γ to uneven chain segment propagation in the chain update in a nonuniform density profile.