Statistical mechanics approach to a general hyperbranched polymer system consisting of ABg monomers and Cf cores

A general hyperbranched polymerization system that consists of ABg monomers and Cf cores is studied by the principle of statistical mechanics. Two types of partition functions for the polymerization are constructed from different viewpoints, and then the explicit expressions of equilibrium size distribution function, equilibrium free energy as well as the law of mass action are obtained. To investigate the average properties of the system, the number-, weight-average degree of polymerization and polydispersity index are derived. Especially, to investigate the average spatial dimension of the hyperbranched polymers, the mean square radius of gyration is presented as a focus and the influence of unequal reactivity on which is taken into account. As an application, several thermodynamic properties such as the equation of state, isothermal compressibility and specific heat are calculated. In addition, some structural parameters of hyperbranched polymers are also presented under various conditions. The results show that two types of competitions take place during the growing course of hyperbranched polymers, and the reactivity parameter has significant effect on the competitions, which are closely related to the thermodynamic properties and spatial dimension of the hyperbranched polymers.