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We have examined the critical adsorption transition for a nonreversible random walk chain (NRRW) and a self-avoiding walk chain (SAW) at the surface and have identified the critical adsorption point. We then examine the partition coefficient of a NRRW chain and a SAW chain confined in a slit with two adsorbing walls near the critical adsorption point. For a NRRW chain, the partition coefficient at the critical adsorption point is equal to one and is independent of the chain length. This property has been utilized in the liquid chromatography operated at the critical adsorption point (LCCAP). For a SAW chain, however, the partition coefficient at the critical adsorption point K er is greater than one and is dependent on the chain length and slit width. This confirms the experimental observation of a weak chain length dependence of the partition coefficient at the critical point. Moreover, the study found that K er for a SAW scales with (R g0 /D 2 ) to a first-order approximation where R g0 is the radius of the gyration of the unconfined chain in the dilute solution and D is the slit width.

We present a lattice computer simulation study on the geometrical confinement effect of polymer solutions in a wide range of concentrations. Polymer chains were equilibrated for a system that consists of a bulk solution and a slit between two parallel walls. The partition coefficient of the polymer was determined and found to increase as the bulk concentration Φ increased, confirming the theoretical predictions. We applied the blob picture to analyze the partition coefficient data in the semidilute regime where the partitioning is considered to be controlled by the ratio of the blob size to the slit width, rather than by the ratio of the chain dimension to the slit width. Our simulation data and earlier experimental data support the application of the blob theory. Furthermore, we found it is possible to estimate the blob size in the bulk solution directly from the partition coefficient data by assuming that the confinement entropy of the blob depends on the blob size in the same way as the confinement entropy of the individual chain depends on the chain dimension at infinite dilution. The blob size thus determined confirms the scaling prediction; namely, the blob size ∞ Φ -3/4 . Some deviations from this simple picture, however, exist when the slit is narrow and the solution is concentrated.

We employ computer simulations and thermodynamic integration to analyse the effects of bending rigidity and slit confinement on the free energy cost of tying knots, $\Delta F_{\rm knotting}$, on polymer chains under tension. A tension-dependent, non-zero optimal stiffness $\kappa_{\rm min}$ exists, for which $\Delta F_{\rm knotting}$ is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness $\kappa_{\rm min}$ will be preferred by the knot. A {\it local} analysis of the bending in the interior of the knot reveals that local stretching of chains at the braid region is responsible for the fact that the tension-dependent optimal stiffness has a non-zero value. The reduction in $\Delta F_{\rm knotting}$ for a chain with optimal stiffness relative to the flexible chain can be enhanced by tuning the slit width of the 2D confinement and increasing the knot complexity. The optimal stiffness itself is independent of the knot types we considered, while confinement shifts it towards lower values.