A general, analytically tractable, nonperturbative theory of the equilibrium structure of dense polymer melts is proposed on the basis of modern integral-equation theories of molecular liquids. Calculations are presented for polymer rings obeying Gaussian statistics and interacting via hard-core repulsions. The correlation hole, compressibility, and static structure factor are found to be sensitive functions of liquid density and degree of polymerization.PACS numbers: 61.25.Hq, 61.20.Gy, 61.25.Em, 61.41.+e Theoretical understanding of the structure of polymer solutions has increased dramatically in the past two decades because of the development of scaling, renormalization-group, and self-consistent-field techniques. 1_4 On the other hand, the dense polymeric liquid is characterized by very strong intermolecular interactions and has remained virtually intractable analytically. Moreover, Monte Carlo and molecular-dynamics simulations of both lattice and continuum models of polymer melts are extremely computationally intensive and have generally been limited to small systems composed of relatively short chains. 5 The purpose of this Letter is to present the first systematic, continuum analytic approach to the equilibrium structure of polymer melts based on nonperturbative statistical mechanical theories of molecular fluids. The central result of our work is the calculation of the intermolecular radial distribution function for the polymer liquid. This quantity is of significance not only for structural reasons, but also because it where p is the molecular number density, h, C, and co are square matrices of rank TV (for molecules containing TV interaction sites) with elements h ay (r), C ay (r), and Q) ay (r), respectively. More specifically, h ay (r)=g ay (r) -1, where g a y(r) is the intermolecular site-site radial distribution function, C ay (r) is the corresponding direct correlation function, and co ay (r) is the jTUramolecular probability distribution function for sites a and y on the same molecule and describes chemical connectivity and flexibility. The direct correlation function defined in Eq.(1) can be interpreted 6 physically as an effective pair potential in the liquid and is thermodynamic state and intramolecular structure dependent. The integral equation approach is particularly appealing since the microscopic intermolecular interactions enter explicitly via the direct correlation function. 7 For flexible molecules the intramolecular and intermolecular correlations must be determined self-consistently.Since rapidly varying repulsive forces dominate the dense liquid structure, the problem is generally simplified by adopting a hard-core provides a direct route to the calculation of all the thermodynamic properties, including an equation of state beyond the artificial lattice or cell models.The modern theory of one-component atomic and small-molecule liquids has advanced to a mature stage. 6,7 In the absence of strong attractive forces, short-range order is controlled by the harsh repulsive forces wh...
A molecular interpretation of the viscoelastic behavior of elastomers at long times is developed. This interpretation is based on a model consisting of dangling chain ends (branches) in a cross-linked network with topological constraints (entanglements). We make use of results by de Gennes for the reptation of a single branched chain with topological constraints. We then sum over contributions to the relaxation modulus from a distribution of branch lengths. The distribution function for chain branches was obtained by assuming that the network was formed from primary molecules due to a random cross-linking process. This leads to a relaxation modulus having power law dependence on time as in the empirical Chasset and Thirion equation. The theory also predicts that the parameters in the Chasset and Thirion equation have a cross-link density dependence which is consistent with available experimental data on natural rubber. Furthermore, the present treatment predicts that relaxation curves of different cross-link density are approximately superposable along the log time axis. The shift factor is predicted to have power law dependence on cross-link density, as observed experimentally by Plazek. Finally, the present theory predicts that the Plazek exponent (x) for the shift factor is approximately related to the Chasset and Thirion exponent (m) for the relaxation modulus (x ~2/m) in a manner consistent with available experimental data.
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