An Integral Equation Approach to Effective Interactions between Polymers in Solution

Effects of Knots on Ring Polymers in Solvents of Varying Quality

We employ extensive computer simulations to investigate the conformations and the interactions of ring polymers under conditions of worsening solvent quality, in comparison with those for linear polymers. We determine the dependence of the Θ-temperature on knotedness by considering ring polymers of different topologies. We establish a clear decrease of the former upon changing the topology of the polymer from linear to an unknotted ring and a further decrease of the same upon introducing trefoil- or 5-fold knots but we find no difference in the Θ-point between the two knotted molecules. Our results are based on two independent methods: one considering the scaling of the gyration radius with molecular weight and one based on the dependence of the effective interaction on solvent quality. In addition, we calculate several shape-parameters of the polymers to characterize linear, unknotted, and knotted topologies in good solvents and in the proximity of the Θ-point. The shape parameters of the knotted molecules show an interesting crossover at a degree of polymerization that depends on the degree of knottedness of the molecule.

Thermodynamic consistency in variable-level coarse graining of polymeric liquids

Numerically optimized reduced descriptions of macromolecular liquids often present thermodynamic inconsistency with atomistic level descriptions even if the total correlation function, i.e. the structure, appears to be in agreement. An analytical expression for the effective potential between a pair of coarse-grained units is derived starting from the first-principles Ornstein-Zernike equation, for a polymer liquid where each chain is represented as a collection of interpenetrating blobs, with a variable number of blobs, n(b), of size N(b). The potential is characterized by a long tail, slowly decaying with characteristic scaling exponent of N(b)(1/4). This general result applies to any coarse-grained model of polymer melts with units larger than the persistence length, highlighting the importance of the long, repulsive, potential tail for the model to correctly predict both structural and thermodynamic properties of the macromolecular liquid.

Modeling of aqueous poly(oxyethylene) solutions. 2. Mesoscale simulations

We extend our work on aqueous solutions of poly(oxyethylene) oligomers H-(CH2-O-CH2)n -H (POEn). On the basis of atomistic simulations of trimer and decamer solutions (first part of this series of papers), different sets of coarse-grained implicit-solvent potentials have been constructed using the iterative Boltzmann inversion technique. The comparison of structures obtained from coarse-grained simulations (gyration radii, end-to-end distances, radial distribution functions) with atomistic reference simulations and experiments shows that the state-specific potentials are transferable both to a wide concentration range, if the same molecule size is considered, and to at least 2 orders of magnitude larger molecules (in terms of molecular mass). Comparing the performance of different mesoscale potentials, we find different applicability ranges in terms of molecule sizes. The experimental gyration radii for chains comprising up to 1500 monomers are reproduced almost quantitatively by the decamer-fitted potentials with dihedral interactions included. The trimer-fitted potentials reproduce experimental chain dimensions of up to some hundred monomers but seem to become metastable beyond a certain chain length, as we evidenced some chain collapses. Relaxation of large-scale features is 1-2 orders of magnitude faster in the mesoscale simulations than in the atomistic simulations. The diffusion behavior in dependence of concentration is captured correctly when the decamer potential is applied to the decamer itself. For all other chain lengths, we find that time mapping from coarse-grained to atomistic trajectories has to be determined separately for each concentration. Overall, diffusion is 1-2 orders of magnitude faster on the mesoscale, depending considerably on the Lowe-Andersen thermostat parameters. The CG simulations provide an overall speed-up of about 3 orders of magnitude.

Theoretical coarse-graining approach to bridge length scales in diblock copolymer liquids

A microscopic theory for coarse graining diblock copolymers into dumbbells of interacting soft colloidal particles has been developed based on the solution of liquid-state integral equations. The Ornstein-Zernike equation is solved to provide a mesoscopic description of the diblock copolymer system at the level of block centers of mass, and at the level of polymer centers of mass. Analytical forms of the total correlation functions for block-block, block-monomer, and center-of-mass pairs are obtained for a liquid of structurally symmetric diblock copolymers as a function of temperature, density, chain length, and chain composition. The theory correctly predicts thermodynamically driven segregation of diblocks into microdomains as a function of temperature (chi parameter). The coarse-grained description contains contributions from density and concentration fluctuations, with the latter becoming dominant as temperature decreases. Numerical calculations for the block coarse-grained total correlation functions, as a function of the proximity of the system to its phase transition, are presented. Comparison with united atom molecular dynamics simulations are carried out in the athermal regime, where simulations and theory quantitatively agree with no need of adjustable parameters.

Analytical coarse-grained description for polymer melts

Starting from the Ornstein-Zernike equation the authors derive an analytical theory, at the level of pair correlation functions, which coarse grains polymer melts into liquids of interacting soft colloidal particles. Since it is analytical, the presented coarse-graining approach will be useful in developing multiscale modeling procedures to simulate complex fluids of macromolecules. The accuracy of the theory is tested by its capacity to reproduce the liquid structure, as given by the center-of-mass intermolecular total pair correlation function. The theory is found to agree well with the structure predicted by molecular dynamics simulations of the liquid described at the united atom level as well as by molecular dynamics simulations of the liquid of interacting colloidal particles. The authors perform simulations of the liquid of interacting colloidal particles having as input the potential obtained from their analytical total pair correlation function by enforcing the hypernetted-chain closure approximation. Tests systems are polyethylene melts of chains with increasing degrees of polymerization and polymer melts of chains with different chemical architectures. They also discuss the effect of adopting different conventional approximations for intra- and intermolecular monomer structure factors on the accuracy of the coarse-graining procedure, as well as the relevance of higher-order corrections to their expression.

A functional-integral formulation for polymer colloids: Pagonabarraga-Cates free energy revisited

We have formulated a new functional-integral representation with respect to both polymer and monomer densities, along the lines of the current picture treating polymers as soft colloids. Comparison between the resulting form and a model free energy functional of Pagonabarraga and Cates (PC 2001 Europhys. Lett. 55 348) indicates that the PC relation between monomer and polymer concentrations is to be modified, and that further insertion of the order parameter defined by the square of monomer density is indispensable for regularizing a divergent term absent in the PC functional. Moreover, the saddle-point approximation to our functional integral leads to a self-consistent equation which efficiently preincludes the minimum of the Flory-Huggins-type local free energy as input.

Corrections to scaling and crossover from good- to θ-solvent regimes of interacting polymers

We exploit known properties of universal ratios, involving the radius of gyration R(g), the second and third virial coefficients B(2) and B(3), and the effective pair potential between the centers of mass of self-avoiding polymer chains with nearest-neighbor attraction, as well as Monte Carlo simulations, to investigate the crossover from good- to theta-solvent regimes of polymers of finite length L. The scaling limit and finite-L corrections to scaling are investigated in the good-solvent case and close to the theta temperature. Detailed interpolation formulas are derived from Monte Carlo data and results for the Edwards two-parameter model, providing estimates of universal ratios as functions of the observable ratio A(2)=B(2)/R(g) (3) over the whole temperature range, from the theta point to the good-solvent regime. The convergence with L (L< or =8000) is found to be satisfactory under good-solvent conditions, but longer chains would be required to match theoretical predictions near the theta point, due to logarithmic corrections. A quantitative estimate of the universal ratio A(3)=B(3)/R(g) (6) as a function of temperature shows that the third virial coefficient remains positive throughout, and goes through a pronounced minimum at the theta temperature, which goes to zero as 1/ln L in the scaling limit.

Coarse-grained description of polymer blends as interacting soft-colloidal particles

We present a theoretical approach which maps polymer blends onto mixtures of soft-colloidal particles. The analytical mesoscale pair correlation functions reproduce well data from united atom molecular dynamics simulations of polyolefin mixtures without fitting parameters. The theory exactly recovers the analytical expressions for density and concentration fluctuation structure factors of soft-colloidal mixtures (liquid alloys).