^{1}, Chi C. Hua

^{2,a)}and Show A. Chen

^{3}

### Abstract

We propose an ellipsoid-chain model which may be routinely parameterized to capture large-scale properties of semiflexible, amphiphilic conjugated polymers in various solvent media. The model naturally utilizes the defect locations as pivotal centers connecting adjacent ellipsoids (each currently representing ten monomer units), and a variant umbrella-sampling scheme is employed to construct the potentials of mean force (PMF) for specific solvent media using atomistic dynamics data and simplex optimization. The performances, both efficacy and efficiency, of the model are thoroughly evaluated by comparing the simulation results on long, single-chain (i.e., 300-mer) structures with those from two existing, finer-grained models for a standard conjugated polymer (i.e., poly(2-methoxy-5-(2^{′}-ethylhexyloxy)-1,4-phenylenevinylene) or MEH-PPV) in two distinct solvents (i.e., chloroform or toluene) as well as a hybrid, binary-solvent medium (i.e., chloroform/toluene = 1:1 in number density). The coarse-grained Monte Carlo (CGMC) simulation of the ellipsoid-chain model is shown to be the most efficient—about 300 times faster than the coarse-grained molecular dynamics (CGMD) simulation of the finest CG model that employs explicit solvents—in capturing elementary single-chain structures for both single-solvent media, and is a few times faster than the coarse-grained Langevin dynamics (CGLD) simulation of another implicit-solvent polymermodel with a slightly greater coarse-graining level than in the CGMD simulation. For the binary-solvent system considered, however, both of the two implicit-solvent schemes (i.e., CGMC and CGLD) fail to capture the effects of conspicuous concentration fluctuations near the polymer-solvent interface, arising from a pronounced coupling between the solvent molecules and different parts of the polymer. Essential physical implications are elaborated on the success as well as the failure of the two implicit-solvent CG schemes under varying solvent conditions. Within the ellipsoid-chain model, the impact of synthesized defects on local segmental ordering as well as bulk chain conformation is also scrutinized, and essential consequences in practical applications discussed. In future perspectives, we remark on strategy that takes advantage of the coordination among various CG models and simulation schemes to warrant computational efficiency and accuracy, with the anticipated capability of simulating larger-scale, many-chain aggregate systems.

The authors thank the reviewers’ comments leading to a general improvement of this article. This work is supported by the National Science Council of the ROC. The resource provided by the National Center for High-Performance Computing is gratefully acknowledged.

I. INTRODUCTION

II. SIMULATION PROTOCOL

A. Background

B. The ellipsoid-chain model and Monte Carlo scheme

III. RESULTS AND DISCUSSION

A. Single-chain structures in various solvent media

B. Effects of synthesized defect

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Polymers
- 84.0
- Solvents
- 84.0
- Solution polymerization
- 17.0
- Polymer structure
- 15.0
- Brownian dynamics
- 10.0

## Figures

Sketches illustrating (from top to bottom) AMD, CGMD, CGLD, and CGMC realizations of a single MEH-PPV (poly(2-methoxy-5-(2^{′}-ethylhexyloxy)-1,4-phenylenevinylene)) chain in various model descriptions.

Sketches illustrating (from top to bottom) AMD, CGMD, CGLD, and CGMC realizations of a single MEH-PPV (poly(2-methoxy-5-(2^{′}-ethylhexyloxy)-1,4-phenylenevinylene)) chain in various model descriptions.

Pictorial representations of a pair of 10-mer MEH-PPV in (a) vacuum, (b) toluene bath, and (c) chloroform bath (note that the solvent molecules in cases (b) and (c) were not plotted for clarity), where the PMFs have been computed using AMD simulations (symbols). Also, a comparison is shown with the predictions of the parameterized GB potential (lines) for (d) toluene or (e) chloroform medium. For each of the pair arrangements, the mean separation distance varies in the range of 0–20 Å with a uniform increment of 0.1 Å. At a given distance, AMD simulations (with DREIDING force fields) were performed using Nosé–Hoover thermostat at *T* = 298 K (with a coupling constant of 0.1 ps) for 1 ns, with an integration time step of 1 fs and a cutoff distance of 15 Å for all nonbonding interactions. This amounts to a total of 1000 nearly independent atomistic configurations later used to compute the PMFs at a given oligomer separation. The density of the solution was estimated using Nosé–Hoover *NPT* ensemble (1 atm and 298 K) to be 0.9 and 1.4 g/cm^{3} for toluene and chloroform, respectively, with the total numbers of solvent particles being 5800 and 7500, respectively.

Pictorial representations of a pair of 10-mer MEH-PPV in (a) vacuum, (b) toluene bath, and (c) chloroform bath (note that the solvent molecules in cases (b) and (c) were not plotted for clarity), where the PMFs have been computed using AMD simulations (symbols). Also, a comparison is shown with the predictions of the parameterized GB potential (lines) for (d) toluene or (e) chloroform medium. For each of the pair arrangements, the mean separation distance varies in the range of 0–20 Å with a uniform increment of 0.1 Å. At a given distance, AMD simulations (with DREIDING force fields) were performed using Nosé–Hoover thermostat at *T* = 298 K (with a coupling constant of 0.1 ps) for 1 ns, with an integration time step of 1 fs and a cutoff distance of 15 Å for all nonbonding interactions. This amounts to a total of 1000 nearly independent atomistic configurations later used to compute the PMFs at a given oligomer separation. The density of the solution was estimated using Nosé–Hoover *NPT* ensemble (1 atm and 298 K) to be 0.9 and 1.4 g/cm^{3} for toluene and chloroform, respectively, with the total numbers of solvent particles being 5800 and 7500, respectively.

Comparison of the intrachain RDFs (unnormalized) for two single-solvent media, where M/C and M/T denote the MEH-PPV/chloroform and MEH-PPV/toluene, respectively. The CGMD and CGLD simulations were both executed for a rescaled (AMD) time about 10 ns, with a time step 2 fs for CGMD and 5 fs for CGLD. The CGMC simulation involves totally 300 000 steps.^{31}

Comparison of the intrachain RDFs (unnormalized) for two single-solvent media, where M/C and M/T denote the MEH-PPV/chloroform and MEH-PPV/toluene, respectively. The CGMD and CGLD simulations were both executed for a rescaled (AMD) time about 10 ns, with a time step 2 fs for CGMD and 5 fs for CGLD. The CGMC simulation involves totally 300 000 steps.^{31}

The predicted scaling laws for mean end-to-end distance (ETE) of MEH-PPV in (a) chloroform, (b) toluene, and (c) chloroform/toluene = 1:1 in number density. In all cases, an ensemble average based on 100 independent chains was utilized to create the plot, and the symbols represent CGMD (triangles), CGLD (squares), and CGMC (circles). For clarity, the curves have been shifted in the *x*-axis direction by varying degree. The error bar given for the slope of each curve, which yields the solvent quality exponent, was evaluated using the mean values of ETE for the three chain lengths investigated. In this way, it is evident that only the result shown in (c) for the CGMD simulation exhibited appreciable deviation from the other two cases.

The predicted scaling laws for mean end-to-end distance (ETE) of MEH-PPV in (a) chloroform, (b) toluene, and (c) chloroform/toluene = 1:1 in number density. In all cases, an ensemble average based on 100 independent chains was utilized to create the plot, and the symbols represent CGMD (triangles), CGLD (squares), and CGMC (circles). For clarity, the curves have been shifted in the *x*-axis direction by varying degree. The error bar given for the slope of each curve, which yields the solvent quality exponent, was evaluated using the mean values of ETE for the three chain lengths investigated. In this way, it is evident that only the result shown in (c) for the CGMD simulation exhibited appreciable deviation from the other two cases.

Snapshots of single-chain (300-mer MEH-PPV) conformation with various amounts of defect: (a) 10%, (b) 5%, or (c) 0% in toluene; (d) 10%, (e) 5%, or (f) 0% in chloroform. The CGMC simulations were performed using *NVT* (298 K) ensemble for 300 000 steps. Understanding the real impact of defects requires a more detailed analysis; see discussion in the main text and results gathered in Table IV.

Snapshots of single-chain (300-mer MEH-PPV) conformation with various amounts of defect: (a) 10%, (b) 5%, or (c) 0% in toluene; (d) 10%, (e) 5%, or (f) 0% in chloroform. The CGMC simulations were performed using *NVT* (298 K) ensemble for 300 000 steps. Understanding the real impact of defects requires a more detailed analysis; see discussion in the main text and results gathered in Table IV.

## Tables

Intramolecular CG potentials and parameters used for defect locations for the ellipsoid-chain model. The results for the other two CG models were different from what given here due to a different coarse-graining level, and some information may be found in Ref. 4.

Intramolecular CG potentials and parameters used for defect locations for the ellipsoid-chain model. The results for the other two CG models were different from what given here due to a different coarse-graining level, and some information may be found in Ref. 4.

Parameter values for the intermolecular GB potential determined for three solvent media, each ellipsoid segment representing ten monomer units of MEH-PPV. The GB potential utilized is given by , where **u** _{ i } and **u** _{ j } are the unit vectors describing the orientations of the ellipsoid pairs *i* and *j* under consideration, **r** _{ ij } is a vector connecting their geometric centers, σ_{0} is related to the ellipsoidal dimension, and ξ is a dimensionless parameter which helps control the width of the potential well independent of its depth or location of the potential minimum. The parameters χ and (ɛ_{0}, χ^{′}, ν, μ) appear in the functionals of σ(**u** _{ i }, **u** _{ j }, **r** _{ ij }) and ɛ(**u** _{ i }, **u** _{ j }, **r** _{ ij }), respectively; see detailed specifications in Ref. 27.

Parameter values for the intermolecular GB potential determined for three solvent media, each ellipsoid segment representing ten monomer units of MEH-PPV. The GB potential utilized is given by , where **u** _{ i } and **u** _{ j } are the unit vectors describing the orientations of the ellipsoid pairs *i* and *j* under consideration, **r** _{ ij } is a vector connecting their geometric centers, σ_{0} is related to the ellipsoidal dimension, and ξ is a dimensionless parameter which helps control the width of the potential well independent of its depth or location of the potential minimum. The parameters χ and (ɛ_{0}, χ^{′}, ν, μ) appear in the functionals of σ(**u** _{ i }, **u** _{ j }, **r** _{ ij }) and ɛ(**u** _{ i }, **u** _{ j }, **r** _{ ij }), respectively; see detailed specifications in Ref. 27.

Comparison of computing times for three CG models (based on a single CPU for an 300-mer MEH-PPV).

Comparison of computing times for three CG models (based on a single CPU for an 300-mer MEH-PPV).

Single-chain structures predicted by CGMC simulation of the ellipsoid-chain model with varying degree of defect.

Single-chain structures predicted by CGMC simulation of the ellipsoid-chain model with varying degree of defect.

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