Integral-equation theory of the structure of polymer melts

Abstract

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...

Polymer-Mode-Coupling Theory of Finite-Size-Fluctuation Effects in Entangled Solutions, Melts, and Gels. 2. Comparison with Experiment

“…It is probably smaller, perhaps by a factor of 2 or 3. This latter estimate follows from the thread PRISM based connection between the density screening length and the dimensionless measure of long wavelength density fluctuations: 44,45,[51][52][53]55 2 , where κ is the isothermal compressibility and σ the statistical segment length. From scattering experiments or thermodynamic measurements on dense polymer liquids, one finds that S 0 is typically in the range 0.15-0.3.…”

“…The packing length rigorously emerges as essentially equal to F ) (π/3)p from the simplest analytic version of PRISM theory based on the idealized Gaussian thread or string models. 44,45,[51][52][53]55 In any case, for polystyrene an upper bound for the parameter Figures 1-4, 7, and 8. δ ) F rh/b ≈ 0.05 in the melt can be estimated from the experimental data. It is probably smaller, perhaps by a factor of 2 or 3.…”

Intermolecular structure factors of macromolecules in solution: Integral equation results

“…(1), PRISM theory [9,10] explicitly enforces inter-molecular excluded volume by requiring the pair correlation function to vanish for distances smaller than the segment size R.…”

“…Starting with the work of Schweizer and Curro [9], this approach has successfully been extended to macromolecular liquids. The polymer reference interaction site model (PRISM) integral equations have been fruitfully applied to describe inter alia the inter-molecule correlations in dense homopolymer systems, polymer blends and block copolymer melts [10].…”

“…(7) to (11) is a neglect of a special site dependence of the density fluctuations as it might for example arise from chain-end effects for linear polymers [9]. Star polymers, where sites in the core region possibly experience very different local density fluctuations than sites in the star-arms, also would require a more elaborate treatment [10].…”