What is the distance to the wall in lattice simulations?

Lattice Monte Carlo simulations are widely used to study the effect of wall s on the concentration profile in polymer solutions. The scaling theory pre dicts that the monomer density at a distance x froth the wall, reduced by t he bulk density, is proportional to (x/R-g0)(1/t) at low concentrations, an d the correlation length replaces R-g0 in the semidilute solution, where R- g0 is the radius of gyration and v is the Flory exponent for the chain dime nsion. We conducted simulations for long chains on a cubic lattice to find that a positive penetration depth y is needed to see an agreement with the theory. The monomers perceive a theoretical wall at an off-lattice position of y behind the presumed wall on the lattice points. We found y similar to 0.13 of the lattice unit at low concentrations but similar to0.36 in the s emidilute solution for athermal chains. For Theta solutions, y was 0.31-0.3 6 at all concentrations. We ascribe the positive y to uneven chain segment propagation in the chain update in a nonuniform density profile.