Analytical coarse-grained description for polymer melts

Bottom-up Coarse-Graining: Principles and Perspectives

Large-scale computational molecular models provide scientists a means to investigate the effect of microscopic details on emergent mesoscopic behavior. Elucidating the relationship between variations on the molecular scale and macroscopic observable properties facilitates an understanding of the molecular interactions driving the properties of real world materials and complex systems (e.g., those found in biology, chemistry, and materials science). As a result, discovering an explicit, systematic connection between microscopic nature and emergent mesoscopic behavior is a fundamental goal for this type of investigation. The molecular forces critical to driving the behavior of complex heterogeneous systems are often unclear. More problematically, simulations of representative model systems are often prohibitively expensive from both spatial and temporal perspectives, impeding straightforward investigations over possible hypotheses characterizing molecular behavior. While the reduction in resolution of a study, such as moving from an atomistic simulation to that of the resolution of large coarse-grained (CG) groups of atoms, can partially ameliorate the cost of individual simulations, the relationship between the proposed microscopic details and this intermediate resolution is nontrivial and presents new obstacles to study. Small portions of these complex systems can be realistically simulated. Alone, these smaller simulations likely do not provide insight into collectively emergent behavior. However, by proposing that the driving forces in both smaller and larger systems (containing many related copies of the smaller system) have an explicit connection, systematic bottom-up CG techniques can be used to transfer CG hypotheses discovered using a smaller scale system to a larger system of primary interest. The proposed connection between different CG systems is prescribed by (i) the CG representation (mapping) and (ii) the functional form and parameters used to represent the CG energetics, which approximate potentials of mean force (PMFs). As a result, the design of CG methods that facilitate a variety of physically relevant representations, approximations, and force fields is critical to moving the frontier of systematic CG forward. Crucially, the proposed connection between the system used for parametrization and the system of interest is orthogonal to the optimization used to approximate the potential of mean force present in all systematic CG methods. The empirical efficacy of machine learning techniques on a variety of tasks provides strong motivation to consider these approaches for approximating the PMF and analyzing these approximations.

Iterative integral equation methods for structural coarse-graining

In this paper, new Newton and Gauss-Newton methods for iterative coarse-graining based on integral equation theory are evaluated and extended. In these methods, the potential update is calculated from the current and target radial distribution function, similar to iterative Boltzmann inversion, but gives a potential update of quality comparable with inverse Monte Carlo. This works well for the coarse-graining of molecules to single beads, which we demonstrate for water. We also extend the methods to systems that include coarse-grained bonded interactions and examine their convergence behavior. Finally, using the Gauss-Newton method with constraints, we derive a model for single bead methanol in implicit water, which matches the osmotic pressure of the atomistic reference. An implementation of all new methods is provided for the open-source VOTCA package.

Evaluation of mapping schemes for systematic coarse graining of higher alkanes

Different levels of coarse-graining are introduced to study the thermodynamic and structural properties of higher alkanes.

First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids

We present a detailed derivation and testing of our approach to rescale the dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y. Lyubimov, J. McCarty, A. Clark, and M. G. Guenza, J. Chem. Phys. 132, 224903 (2010)). Starting from the first-principle Liouville equation and applying the Mori-Zwanzig projection operator technique, we derive the generalized Langevin equations (GLEs) for the coarse-grained representations of the liquid. The chosen slow variables in the projection operators define the length scale of coarse graining. Each polymer is represented at two levels of coarse graining: monomeric as a bead-and-spring model and molecular as a soft colloid. In the long-time regime where the center-of-mass follows Brownian motion and the internal dynamics is completely relaxed, the two descriptions must be equivalent. By enforcing this formal relation we derive from the GLEs the analytical rescaling factors to be applied to dynamical data in the coarse-grained representation to recover the monomeric description. Change in entropy and change in friction are the two corrections to be accounted for to compensate the effects of coarse graining on the polymer dynamics. The solution of the memory functions in the coarse-grained representations provides the dynamical rescaling of the friction coefficient. The calculation of the internal degrees of freedom provides the correction of the change in entropy due to coarse graining. The resulting rescaling formalism is a function of the coarse-grained model and thermodynamic parameters of the system simulated. The rescaled dynamics obtained from mesoscale simulations of polyethylene, represented as soft-colloidal particles, by applying our rescaling approach shows a good agreement with data of translational diffusion measured experimentally and from simulations. The proposed method is used to predict self-diffusion coefficients of new polyethylene samples.

Use of the Ornstein-Zernike Percus-Yevick equation to extract interaction potentials from pair correlation functions

In this work, we test the ability of the Ornstein-Zernike equation in the Percus-Yevick approximation (OZPY) to generate interaction potentials from pair correlation functions (PCFs) of monatomic and diatomic Lennard-Jones fluids. The PCFs are generated by solving OZPY equation (monatomic fluid) and molecular-dynamics (MD) simulations (diatomic fluid). Since the interaction potentials are inputs in the OZPY method and the MD simulation, the extraction of the potential from the PCFs using OZPY is a test of self-consistency. This test is necessary if the procedure is to be used to generate coarse-grained (CG) potentials from PCFs. We find that the procedure is completely self-consistent for the monatomic fluid in the whole range of densities studied (reduced density up to 0.55, under reduced temperature of 2.0). In the diatomic case, we find that the procedure is generally self-consistent under both low and high densities, although there is a systematic deviation at high densities. The method is able to reproduce the two parameters (ε and σ) of the input Lennard-Jones potential model to within about 1%. This CG potential generating procedure can be straightforwardly extended to more complicated molecules.

Coarse-grained models of tethers for fast self-assembly simulations

Long molecular ligands or "tethers" play an important role in the self-assembly of many nanoscale systems. These tethers, whose only interaction may be a hard-core repulsion, contribute significantly to the free energy of the system because of their large conformational entropy. Here, we investigate how simple approximate models can be developed and used to quickly determine the configurations into which tethers will self assemble in nanoscale systems. We derive criteria that determine when these models are expected to be accurate. Finally, we propose a generalized two-body approximation that can be used as a toy model for the self-assembly of tethers in systems of arbitrary geometry and apply this to the self-assembly of self-assembled monolayers on a planar surface. We compare our results to those in the literature obtained via atomistic and dissipative particle dynamics simulations.

Cooperative dynamics in homopolymer melts: a comparison of theoretical predictions with neutron spin echo experiments

We present a comparison between theoretical predictions of the generalized Langevin equation for cooperative dynamics (CDGLE) and neutron spin echo data of dynamic structure factors for polyethylene melts. Experiments cover an extended range of length and time scales, providing a compelling test for the theoretical approach. Samples investigated include chains with increasing molecular weights undergoing dynamics across the unentangled to entangled transition. Measured center-of-mass (com) mean-square displacements display a crossover from subdiffusive to diffusive dynamics. The generalized Langevin equation for cooperative dynamics relates this anomalous diffusion to the presence of the interpolymer potential, which correlates the dynamics of a group of slowly diffusing molecules in a dynamically heterogeneous liquid. Theoretical predictions of the subdiffusive behavior, of its crossover to free diffusion, and of the number of macromolecules undergoing cooperative motion are in quantitative agreement with experiments.

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.