Molecular design: Mathematical models and computer enumerations of polymers with loops

The mathematical formalisms of two related polymeric models, trails and silhouettes are presented. Trails are walks on regular crystal lattices. They are allowed to intersect their own path, but are forbidden to go more than once along the same bond. Silhouettes are just the shadows of trails. By incorporating a fugacity factor e^@q (@q = -|@e|k"BT, |@e| is the energy of intersection) with each intersection, the thermodynamical properties of these models may be controlled. As such, the models may describe the behaviors of polymers with loops in various solvent conditions. The persistence lengths (defined as the vestiges induced by fixing the direction of the first bond) are systematically studied on two-dimensional square and triangular lattices, and on three-dimensional cubic and face-centered cubic lattices. They are found to scale like ~ l^p^k^@n^(^@q^)@?(l) where p is a parameter, k = 0, 1, 2,..., @n(@q) is the correlation exponent, and f(l) is some function of the chain length l. As the temperature is decreased from @q = -~ to large positive values, the persistence lengths undergo a rapid change in a narrow region of @q. This rapid change is identified with the transition from a swollen phase to a collapse phase of the polymeric configuration. The transition temperature can be located by a derivation of the averaged persistence length with respect to the temperature.