Stanley Windwer

Monte Carlo Generation of a Restricted Random Walk and the Excluded‐Volume Problem

A Monte Carlo procedure was used to simulate non self‐intersecting random walks on a high‐speed digital computer. The specific model considered hard cores of 3½ units and different rotational angles. The mean‐square end‐to‐end distance and the mean‐square radius of gyration were fitted to an equation of the form 〈J N2〉 = aNb. It was found that the results obtained by considering nonlattice systems are not consistent with walks constrained to a lattice. This discrepancy is discussed in terms of intrinsic excluded volume and excess excluded volume. It is suggestive that the ratio 〈S N2〉/〈R N2〉 may be a more general parameter in discussing these systems.

Monte Carlo Study of Flexible Branched Macromolecules

A Monte Carlo procedure was used to simulate flexible branched macromolecules on a high‐speed digital computer. The molecules studied were of the regular‐comb type. Configurational and statistical thermodynamic properties were evaluated according to prescriptions used for linear flexible chains. The effects of branching on these properties are discussed.

Self‐Avoiding Walks on the Tetrahedral Lattice

Self‐avoiding walks on the tetrahedral lattice were studied. By extending the excluded‐volume condition to incorporate first nonbonded nearest neighbors one obtains a value of the polymer index γ to be 1.255. This is in agreement with the direct enumeration studies of Mazur and Joseph and is interpreted in terms of the intrinsic excluded volume and excess excluded volume. It is suggested that all real systems may have to be studied in their own right to obtain their molecular‐weight configurational dependence.

Monte Carlo study of polyelectrolyte behavior. I. Technique

A dynamic Monte Carlo method for the relaxation of a polyion in the presence of its counter ions is reported. A lattice model incorporating long range interaction of a Coulombic and excluded volume nature is considered. Results are listed for the configurational energy 〈UNI/NIkT〉, as well as the mean square end‐to‐end distance 〈R2N〉, and the mean square radius of gyration 〈S2N〉, for short chains.

Monte Carlo study of polyelectrolyte behavior. II.

A Monte Carlo simulation of an isolated polyelectrolyte chain in the presence of its counter ions without added salt is reported. Configurational properties such as the mean square end‐to‐end distance 〈R2N〉, the mean square radius of gyration 〈S2N〉, and the mean reduced electrostatic energy 〈UNI/NIkT〉, were studied as a function of the ionic strength of the solution, C; and the % charge of the chain. System behavior is classified into three regions: I: κ−1≪〈S2N〉1/2; II: κ−1∼1/2<S2N〉1/2; III: κ−1≫〈S2N 〉1/2, where κ−1 is the Debye–Huckel radius describing the scale of order of interionic interactions. In region I, the polyelectrolyte behaves configurationally as a random coil and the system is describable electrically as a simple electrolyte. In region II, the polyelectrolyte undergoes a conformational transition and the system electrically deviates notably from simple electrolyte behavior. In region III, the polyelectrolyte behaves configurationally as a rod and the system is characterized electrically by ...

Properties of Self‐Avoiding Walks Not Constrained to Lattices

A study of self‐avoiding walks not confined to a lattice was performed. By use of the method of direct enumeration, walks of 14 steps were generated. Two different models were studied. Configurational properties, such as the mean square end‐to‐end distance, the mean‐square radius of gyration, etc., plus hydrodynamic as well as phase transition properties were obtained. They are discussed in terms of the theories incorporating the excluded volume effect. Some of the results were applied to the DNA molecule.

Monte Carlo Study of Rings on the Tetrahedral Lattice

A Monte Carlo study of the configurational and statistical thermodynamic properties of ring systems generated on the tetrahedral lattice is presented. It is shown that the above parameters obtained are in close agreement with those found for open walks. It is suggested that a number of ratios of configurational parameters may be fruitful quantities for studying a variety of systems.

Study of Self‐Avoiding Walks with Excluded First Nearest Neighbors

A study of the direct enumeration of self‐avoiding walks on the tetrahedral and four choice cubic lattices with first nearest neighbors excluded is presented. Data is given for the mean square end‐to‐end distance, the mean square radius of gyration, the number of returns to the origin, the mean square radius of gyration of the rings, and probability calculations. The results are interpreted in light of the theories of excluded volume.

Erratum: Direct Enumeration Study of Self‐Avoiding Walks on the Tetrahedral Lattice

By application of a general procedure devised by Martin, we generated, up to N = 14, the number of self‐avoiding open chains, and determined their mean‐square end‐to‐end distance, their radius of gyration, the number of returns to the origin, and its corresponding mean‐square end‐to‐end distances. The self‐avoiding chain results were in excellent agreement with Monte Carlo calculations, and the mean‐square radius of gyration of ring systems agreed with our previous Monte Carlo estimates. The number of returns to the origin was used to calculate the order of a phase transition for a tetrahedral model of the helix‐to‐random‐coil system. The higher‐order transition found is the same as that previously obtained by Fisher for other three‐dimensional model systems.

On the topology of loop‐erased self‐avoiding random walks

Loop‐erased self‐avoiding random walks were generated on a 5‐choice cubic lattice. Knot formation in these walks were determined by counting the number of crossings in each walk. The results were fitted to a power law.