A statistical theory for self-condensing vinyl polymerization

Abstract

The method of statistical mechanics is used to investigate several properties of self-condensing vinyl polymerization system. Under the framework of mean field theory, two types of canonical partition functions of the system are constructed from viewpoints of functional groups and polymers, and by which the explicit expressions of equilibrium free energy and the law of mass action are obtained. Based on the free energy, the same size distribution function of hyperbranched polymers is derived by two different methods, which is identical with the previous result given by solving the kinetic differential equations. This implies that the two partition functions are consistent with each other. Furthermore, in terms of the size distribution function, the kth radius of gyration as well as its scaling behavior near the critical point are studied, and the corresponding scaling law is given. As an application, the isothermal compressibility is derived on the basis of equation of state, which indicates that the spatial correlation between monomers increases with the increasing of conversion of the double bonds, and reaches the maximum at the critical point. In addition, it is shown that a usual treatment on the polydispersity index would lead it to infinite, which is not agreement with the true result of 1 at the end of the reaction system. To clarify this fact, we find that the correlation length plays an important role, and then by using asymptotic forms of the size distribution and the second moment, the reasonable result can be carried out.

Statistical thermodynamic properties of a new self-condensing vinyl polymerization system

“…is due to the indistinguishability of i elemental structural units in an i-mer. This result shows that the functionality f makes  i in * AB f become f i1 times of that in an AB * system [25]. The free energy F 2 () corresponding to the partition function Q 2 (N, ) can be expressed as:…”

“…It should be noted that, throughout the paper, the classical assumptions of ideal polymerization have been adopted, namely, (1) all the double bonds of vinyl group A are of equal reactivities; (2) all the double bonds of vinyl group A react independently; and (3) the intramolecular reaction is excluded throughout the paper. In fact, these approximations had been used in previous studies [15][16][17][18][25][26][27][28].…”

“…As a consequence, both the initiating center B * and the newly created propagating center A * can react with the vinyl group of another molecule to form polymers [9,15,25]. For the * AB f system under consideration, symbols of A and B * have the same interpretation as that in AB * inimers, where the index f is the functionality of B * groups.…”

“…Recently, the method of statistical mechanics has been applied to study a series of SCVP systems [25][26][27][28], and the equilibrium free energy and law of mass action were given in their explicit expressions. Furthermore, some average physical quantities such as polymeric moments and mean square radius of gyration can be correlated with thermodynamic quantities of the system.…”

“…To study the related physical and chemical properties of a system with a great number of species, the method of statistical mechanics is often used as a powerful theoretical tool [25][26][27][28]. In this section, two kinds of partition functions would be constructed from different viewpoints, then the equilibrium free energy, law of mass action, and the size distribution function are given in their explicit forms.…”