Winding angles of long lattice walks.

We study the winding angles of random and self-avoiding walks (SAWs) on square and cubic lattices with number of steps N ranging up to 10(7). We show that the mean square winding angle 〈θ(2)〉 of random walks converges to the theoretical form when N → ∞. For self-avoiding walks on the square lattice, we show that the ratio 〈θ(4)〉/〈θ(2)〉(2) converges slowly to the Gaussian value 3. For self-avoiding walks on the cubic lattice, we find that the ratio 〈θ(4)〉/〈θ(2)〉(2) exhibits non-monotonic dependence on N and reaches a maximum of 3.73(1) for N ≈ 10(4). We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of lnN independent segments of the walk, where the ith segment contains 2(i) steps. We find that the square winding angle of the ith segment increases approximately as i(0.5), which leads to an increase of the total square winding angle proportional to (lnN)(1.5).

Long polymers near wedges and cones.

We perform a Monte Carlo study of N-step self-avoiding walks, attached to the corner of an impenetrable wedge in two dimensions (d=2), or the tip of an impenetrable cone in d=3, of sizes ranging up to N=10(6) steps. We find that the critical exponent γ(α), which determines the dependence of the number of available conformations on N for a cone or wedge with opening angle α, is in good agreement with the theory for d=2. We study the end-point distribution of the walks in the allowed space and find similarities to the known behavior of random walks (ideal polymers) in the same geometry. For example, the ratio between the mean square end-to-end distances of a polymer near the cone or wedge and a polymer in free space depends linearly on γ(α), as is known for ideal polymers. We show that the end-point distribution of polymers attached to a wedge does not separate into a product of angular and radial functions, as it does for ideal polymers in the same geometry. The angular dependence of the end position of polymers near the wedge differs from theoretical predictions.

Entropic pressure in lattice models for polymers.

In lattice models, local pressure on a surface is derived from the change in the free energy of the system due to the exclusion of a certain boundary site, while the total force on the surface can be obtained by a similar exclusion of all surface sites. In these definitions, while the total force on the surface of a lattice system matches the force measured in a continuous system, the local pressure does not. Moreover, in a lattice system, the sum of the local pressures is not equal to the total force as is required in a continuous system. The difference is caused by correlation between occupations of surface sites as well as finite displacement of surface elements used in the definition of the pressures and the force. This problem is particularly acute in the studies of entropic pressure of polymers represented by random or self-avoiding walks on a lattice. We propose a modified expression for the local pressure which satisfies the proper relation between the pressure and the total force, and show that for a single ideal polymer in the presence of scale-invariant boundaries it produces quantitatively correct values for continuous systems. The required correction to the pressure is non-local, i.e., it depends on long range correlations between contact points of the polymer and the surface.

Ideal polymers near scale-free surfaces.

The number of allowed configurations of a polymer is reduced by the presence of a repulsive surface resulting in an entropic force between them. We develop a method to calculate the entropic force, and detailed pressure distribution, for long ideal polymers near a scale-free repulsive surface. For infinite polymers the monomer density is related to the electrostatic potential near a conducting surface of a charge placed at the point where the polymer end is held. Pressure of the polymer on the surface is then related to the charge density distribution in the electrostatic problem. We derive explicit expressions for pressure distributions and monomer densities for ideal polymers near a two- or three-dimensional wedge, and for a circular cone in three dimensions. Pressure of the polymer diverges near sharp corners in a manner resembling (but not identical to) the electric field divergence near conducting surfaces. We provide formalism for calculation of all components of the total force in situations without axial symmetry.