Confinement-force scaling of flexible polymers: From Gaussian chains to self-avoiding walks

The confinement of flexible polymers between parallel walls generates entropic forces whose magnitude and scaling depend on excluded-volume strength. While ideal Gaussian and fully self-avoiding walk (SAW) chains represent well-characterized limits, a unified description of confinement forces across the crossover between them has remained incomplete. We develop such a description by expressing confinement free energies and forces in terms of the dimensionless ratio L/Rg, where L is the slit width and Rg is the unconfined radius of gyration. Using the exactly solvable Gaussian chain as an analytic benchmark, validated with sub-percent accuracy by a Brownian-bridge-corrected Rosenbluth estimator, we establish the asymptotic force law f·Rg∼L/Rg−3 and deploy the same framework across the full Gaussian–SAW crossover. For the absorbing-wall calculations, confinement is defined through the survival of the tethered chain within the slit. Monte Carlo simulations of self-avoiding walks yield an approximate collapse, without adjustable parameters, of f·Rg vs L/Rg across chain lengths, with measurable residual chain-length dependence and an effective exponent broadly consistent with the theoretical SAW limit −1/ν+1≈−2.70. Using the Domb–Joyce model to interpolate continuously between the two limits, we test whether the confinement response at fixed w becomes approximately organized across chain lengths when expressed in terms of the independently measured L/Rg, and quantify the remaining finite-chain deviations. Systematic tests across boundary conditions and spatial dimensions show that ideal Gaussian chains have a strictly vanishing confinement force under the discrete fold-back reflecting ensemble used here, whereas finite excluded-volume interactions generate a measurable force under the same ensemble. These results demonstrate the sensitivity of the confinement response to both polymer interactions and the statistical definition of the boundary condition. Across all conditions examined, the boundary condition produces a systematic change in the force amplitude and detailed scaling response, while the leading organization by Rg is retained as a leading-order description. However, the reflecting-to-absorbing force ratio varies with the reduced slit width and the residuals are systematic, so the boundary condition is not a purely multiplicative renormalization of the force amplitude, and the data do not establish an exact universal master curve. These results show that the diversity of confinement responses across polymer models, interaction strengths, and boundary conditions is organized at leading order by size scaling via Rg, providing a useful scaling description of flexible-polymer confinement, with measurable finite-size, interaction-dependent, and boundary-dependent corrections, relevant to soft-matter and biophysical systems.