Probabilistic-topological representation for crosslink, entanglement, or contact networks in (bio)polymer systems

Structure-property relationships for (bio)polymer systems, especially for crosslinked network systems, are discussed and calculated. Physical properties of the system, e.g., modulus of elasticity, gel fraction, etc., are expressed as certain functionals of the system’s actual structure. A new topological gelation criterion for the sol-gel transition point is proposed. For the first time the gel fraction for a system with closed cycles is calculated, using a new method of equivalent crosslinks. In contrast to many existing theories, the present theory may easily be applied to crosslinking systems far from thermodynamic equilibrium and to nonrandom systems. It is suggested that a new kind of auto-oscillatory phenomenon, namely an oscillating sol-gel transition, is to be found in open crosslinked systems, e.g., in actomyosin solutions maintained far from equilibrium by a constant influx of ATP. Such phenomena may be of great importance in living cells. The theory is easily adaptable to a broad spectrum of systems to which probabilistic-topological notions similar to the ones introduced here may be applied, e.g., to entanglement networks for systems of flexible chains or to contact networks for systems of stiff molecules.