Excluded volume

Abstract: The general relationship between the percolation threshold of systems of various objects and the excluded volume associated with these objects is discussed. In particular, we derive the average excluded area and the average excluded volume associated with two- and three-dimensional randomly oriented objects. The results yield predictions for the dependencies, of the percolation critical concentration of various kinds of "sticks," on the stick aspect ratio and the anisotropy of the stick orientation distribution. Comparison of the present results with available Monte Carlo data shows that the percolation threshold of the sticks is described by the above dependencies. On the other hand, the numerical values of the excluded area and the excluded volume are not dimensional invariants as suggested in the literature, but rather depend on the randomness of the stick orientations. The usefulness of the present results for percolation-threshold problems in the continuum is discussed. In particular, it is shown that the excluded area and the excluded volume give the number of bonds per object ${B}_{c}$ when the objects are all the same size. In the case where there is a distribution of object sizes, the proper average of the excluded area or volume is a dimensional invariant while ${B}_{c}$ is not.

Abstract: The present state of the theory of volume effects in a polymer chain is reviewed. The theory of coils (i.e., chains with predominately repulsive volume interaction of monomers) and the analogy between the excluded volume problem and the theory of second-order phase transitions are briefly described. The theory of globules, formed by attractive interaction of monomers or by external attractive fields, is considered in greater detail. The coil-globule transition under various conditions is analyzed. A theory is constructed for the simplest model of the polymer chain---the model of "interacting beads on a flexible string." The connection of this model with more realistic ones is discussed. This review is based on the approach proposed by I. M. Lifshitz in 1968. While attention is focused on the physical aspects of the problems, some questions concerning biological applications are discussed in the conclusion.

Abstract: A lattice model for dense polymer solutions and polymer mixtures in three dimensions is presented, aiming to develop a model suitable for efficient computer simulation on vector processors, with a qualitatively realistic local dynamics. It is shown that the bond fluctuation algorithm for a suitable set of allowed bond vectors has the property that due to the excluded volume constraint no crossing of bonds by local motions can occur, and entanglement restrictions thus are fully taken into account. For athermal binary (AB) symmetrical polymer mixtures, the dependence of both self‐diffusion coefficient and interdiffusion coefficient on polymer density is obtained, simulating a thin film geometry where a film of polymer A is coated with a film of polymer B. For one density, the dependence of the interdiffusion coefficient on an attractive energy between unlike monomers is also studied. For weak attraction an enhancement of interdiffusion proportional to this energy occurs. For strong attraction, however, a rather immobile tightly bound AB layer forms in the interface which hampers further unmixing.