^{1,a)}and Yuhua Yin

^{1,b)}

### Abstract

The basic idea of fast off-lattice Monte Carlo (FOMC) simulations is to use “soft” repulsive potentials that allow particle overlapping in continuum Monte Carlo(MC) simulations. For multichain systems, this gives much faster chain relaxation and better sampling of the configurational space than conventional molecular simulations using “hard” excluded-volume interactions that prevent particle overlapping. Such coarse-grained models are particularly suitable for the study of equilibrium properties of soft materials. Since soft potentials are commonly used in polymerfield theories, it is another advantage of FOMC simulations that using the *same* Hamiltonian in both FOMC simulations and the theories enables quantitative comparisons between them *without any parameter fitting* to unambiguously reveal the consequences of approximations in the theories. Moreover, FOMC simulations can be performed with various chain models and in any statistical ensemble, and all the advanced off-lattice MC techniques proposed to date can be implemented to further improve the sampling efficiency. We have performed canonical-ensemble FOMC simulations with an isotropic soft pair potential for three systems: we first used (small-molecule) soft spheres to demonstrate the improved sampling of FOMC simulations over conventional molecular simulations; we then used single-chain simulations to show that the effects of excluded-volume interactions can be well captured by the soft repulsive potential; finally, for compressible homopolymer melts, we compared FOMC results with those under the random-phase approximation to demonstrate that FOMC simulations can be used to unambiguously and *quantitatively* reveal the fluctuation/correlation effects in the system. In addition, we examined in detail in our single-chain simulations the spatial discretization scheme used in all previous FOMC simulations.

Q.W. thanks Professor An-Chang Shi for helpful discussion regarding the structure factors. Acknowledgment is made to the donors of The American Chemical Society Petroleum Research Fund for partial support of this research. This work was also supported by the U.S. Department of Energy under Award No. DE-FG02-07ER46448 and by Colorado State University, which are gratefully acknowledged.

I. INTRODUCTION

II. SIMULATION METHODOLOGY

A. System I: Soft spheres

B. System II: Single chain of soft spheres

C. System III: Compressible homopolymer melts

III. RESULTS AND DISCUSSION

A. System I: Soft spheres

B. System II: Single chain of soft spheres

C. System III: Compressible homopolymer melts

IV. CONCLUSIONS

### Key Topics

- Field theory models
- 21.0
- Polymers
- 21.0
- Monte Carlo methods
- 18.0
- Mean field theory
- 17.0
- Spatial analysis
- 13.0

## Figures

FOMC simulations of 256 soft spheres interacting with a soft repulsion given in Eq. (6) at a packing fraction . (a) The ensemble-averaged excess pressure and the number of overlapping particle pairs . For the two cases where (i.e., ), there was no particle overlapping during the course of our simulations. (b) The statistical inefficiency computed from the data and the maximum displacement obtained at an acceptance rate of 93% for trial moves of random particle displacement. measures, on average, the of number Monte Carlo steps needed to generate one statistically uncorrelated sample in the simulations, and larger corresponds to statistically less correlated samples collected in the simulations. In both plots, the ideal-gas (“IG”, where ) case is shown on the left axis, while the hard-sphere (“HS”, where ) case is on the right axis.

FOMC simulations of 256 soft spheres interacting with a soft repulsion given in Eq. (6) at a packing fraction . (a) The ensemble-averaged excess pressure and the number of overlapping particle pairs . For the two cases where (i.e., ), there was no particle overlapping during the course of our simulations. (b) The statistical inefficiency computed from the data and the maximum displacement obtained at an acceptance rate of 93% for trial moves of random particle displacement. measures, on average, the of number Monte Carlo steps needed to generate one statistically uncorrelated sample in the simulations, and larger corresponds to statistically less correlated samples collected in the simulations. In both plots, the ideal-gas (“IG”, where ) case is shown on the left axis, while the hard-sphere (“HS”, where ) case is on the right axis.

log-log plots of the mean-square chain radius of gyration and end-to-end distance with the number of bonds , obtained from single-chain FOMC simulations using (a) Eq. (6), and (b) and with either Eq. (6) (“no grids”) or an anisotropic and position-dependent pair potential. In this latter case, the space is divided into cubic cells of size by grids that are either fixed (“fixed grids”) or randomly shifted after each trial move (“shifted grids”). “HS” in (a) denotes the single-chain FOMC simulations with . In each case, the straight line through symbols represents the unweighed least-squares fit using data points for , with the scaling exponent given in the legend. The straight line without symbol represents the commonly accepted exponent in the long-chain limit.

log-log plots of the mean-square chain radius of gyration and end-to-end distance with the number of bonds , obtained from single-chain FOMC simulations using (a) Eq. (6), and (b) and with either Eq. (6) (“no grids”) or an anisotropic and position-dependent pair potential. In this latter case, the space is divided into cubic cells of size by grids that are either fixed (“fixed grids”) or randomly shifted after each trial move (“shifted grids”). “HS” in (a) denotes the single-chain FOMC simulations with . In each case, the straight line through symbols represents the unweighed least-squares fit using data points for , with the scaling exponent given in the legend. The straight line without symbol represents the commonly accepted exponent in the long-chain limit.

(a) log-log plot of the ensemble-averaged nonbonded interaction energy with the number of bonds , (b) semilogarithmic plot of the average bond length with , and (c) semilogarithmic plot of the single-chain structure factor for , obtained from single-chain FOMC simulations. “DGC” in (c) denotes the ideal case of discrete Gaussian chain with . Refer to the caption of Fig. 2 for more details.

(a) log-log plot of the ensemble-averaged nonbonded interaction energy with the number of bonds , (b) semilogarithmic plot of the average bond length with , and (c) semilogarithmic plot of the single-chain structure factor for , obtained from single-chain FOMC simulations. “DGC” in (c) denotes the ideal case of discrete Gaussian chain with . Refer to the caption of Fig. 2 for more details.

Semilogarithmic plots of the mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length : (a) and , where the filled symbols are obtained with and the open ones with . (b) , , and . In both (c) and (d), , , and .

Semilogarithmic plots of the mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length : (a) and , where the filled symbols are obtained with and the open ones with . (b) , , and . In both (c) and (d), , , and .

(a) Single-chain structure factor and (b) the total structure factor obtained in FOMC simulations of compressible homopolymer melts with and . “DGC” in (a) denotes the ideal case of discrete Gaussian chains (i.e., ), and “RPA” in (b) represents the prediction under the random phase approximation with . Also in (b) each symbol at takes the value of ; see Table I for the corresponding number of chains in each case.

(a) Single-chain structure factor and (b) the total structure factor obtained in FOMC simulations of compressible homopolymer melts with and . “DGC” in (a) denotes the ideal case of discrete Gaussian chains (i.e., ), and “RPA” in (b) represents the prediction under the random phase approximation with . Also in (b) each symbol at takes the value of ; see Table I for the corresponding number of chains in each case.

Effects of in the DGC model on the value of (a) and (b) bulk lamellar period (in units of ) at the ODT of symmetric diblock copolymers obtained under the RPA. Different symbols correspond to the different functional forms of given in the text, with denoting the interaction range.

Effects of in the DGC model on the value of (a) and (b) bulk lamellar period (in units of ) at the ODT of symmetric diblock copolymers obtained under the RPA. Different symbols correspond to the different functional forms of given in the text, with denoting the interaction range.

## Tables

The mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length at various dimensionless chain number density . The case corresponds to the single-chain simulation without the PBCs. Note that we set and the dimensionless excluded-volume parameter .

The mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length at various dimensionless chain number density . The case corresponds to the single-chain simulation without the PBCs. Note that we set and the dimensionless excluded-volume parameter .

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