^{1,a)}, Andrea Pelissetto

^{2,b)}and Carlo Pierleoni

^{3,c)}

### Abstract

We consider a coarse-grained model in which polymers under good-solvent conditions are represented by soft spheres whose radii, which should be identified with the polymer radii of gyrations, are allowed to fluctuate. The corresponding pair potential depends on the sphere radii. This model is a single-sphere version of the one proposed in Vettorel *et al.* [Soft Matter6, 2282 (2010)]10.1039/b921159d, and it is sufficiently simple to allow us to determine all potentials accurately from full-monomer simulations of two isolated polymers (zero-density potentials). We find that in the dilute regime (which is the expected validity range of single-sphere coarse-grained models based on zero-density potentials) this model correctly reproduces the density dependence of the radius of gyration. However, for the thermodynamics and the intermolecular structure, the model is largely equivalent to the simpler one in which the sphere radii are fixed to the average value of the radius of gyration and radii-independent potentials are used: for the thermodynamics there is no advantage in considering a fluctuating sphere size.

C.P. is supported by the Italian Institute of Technology (IIT) under the SEED project Grant No. 259 SIMBEDD Advanced Computational Methods for Biophysics, Drug Design and Energy Research.

I. INTRODUCTION

II. THE MODELS

III. COMPARISON OF THE MODELS

A. Three-body interactions at zero density

B. The semidilute regime

IV. CONCLUSIONS

### Key Topics

- Polymers
- 98.0
- Cumulative distribution functions
- 5.0
- Particle distribution functions
- 5.0
- Polymer structure
- 4.0
- Intermolecular potentials
- 2.0

## Figures

On top we report the center-of-mass potentials β*V*(σ_{1}, σ_{2}; *b*): on the left we show a three-dimensional plot in terms of σ = σ_{1} = σ_{2} and , on the right we show the potentials for σ_{1} = 1.024 and several values of σ_{2}, as a function of *b*. On bottom we show the same plots for the midpoint potentials *V* _{ MP }(σ_{1}, σ_{2}; *b*).

On top we report the center-of-mass potentials β*V*(σ_{1}, σ_{2}; *b*): on the left we show a three-dimensional plot in terms of σ = σ_{1} = σ_{2} and , on the right we show the potentials for σ_{1} = 1.024 and several values of σ_{2}, as a function of *b*. On bottom we show the same plots for the midpoint potentials *V* _{ MP }(σ_{1}, σ_{2}; *b*).

Rescaled potentials as a function of for several values of σ_{1} and σ_{2}. We also plot the function , with ε = 4.42, α = 1.42 (VBK).

Rescaled potentials as a function of for several values of σ_{1} and σ_{2}. We also plot the function , with ε = 4.42, α = 1.42 (VBK).

Three-body potential of mean force β*V* _{3}(**r** _{12}, **r** _{13}, **r** _{23}) for *r* _{12} = *r* _{13} = *r* _{23} = *r*, as a function of . On the left we report results for models M2a, M2c, for the tetramer model (t) of Ref. 22, and the predictions of full-monomer simulations for the quantity associated with the center of mass (FMa); on the right we report the results for model M2b and the prediction of full-monomer simulations for the analogous quantity associated with the polymer midpoint (FMb).

Three-body potential of mean force β*V* _{3}(**r** _{12}, **r** _{13}, **r** _{23}) for *r* _{12} = *r* _{13} = *r* _{23} = *r*, as a function of . On the left we report results for models M2a, M2c, for the tetramer model (t) of Ref. 22, and the predictions of full-monomer simulations for the quantity associated with the center of mass (FMa); on the right we report the results for model M2b and the prediction of full-monomer simulations for the analogous quantity associated with the polymer midpoint (FMb).

Function *F* _{3}(*b*) as a function of *b*, for polymers (FMa), for model M2a, and for the tetramer model of Ref. 22 (t).

Function *F* _{3}(*b*) as a function of *b*, for polymers (FMa), for model M2a, and for the tetramer model of Ref. 22 (t).

Compressibility factor *Z* as a function of Φ. On the left we report results for models M1a, M2a, and M2c, on the right we report the results for models M1b and M2b. They are compared with the polymer prediction *Z* _{ FM } (full line, FM) (from Ref. 41). In the insets we report the deviations 100(*Z*/*Z* _{ FM } − 1).

Compressibility factor *Z* as a function of Φ. On the left we report results for models M1a, M2a, and M2c, on the right we report the results for models M1b and M2b. They are compared with the polymer prediction *Z* _{ FM } (full line, FM) (from Ref. 41). In the insets we report the deviations 100(*Z*/*Z* _{ FM } − 1).

Intermolecular distribution function for several models at Φ = 1.09 and 4.36 (the corresponding function is shifted upward for clarity). On the left we report results for models M1a, M2a, M2c, and the polymer center-of-mass distribution from full-monomer simulations (FMa); on the right we report the results for models M1b, M2b, and the polymer distribution function associated with the polymer midpoint (FMb).

Intermolecular distribution function for several models at Φ = 1.09 and 4.36 (the corresponding function is shifted upward for clarity). On the left we report results for models M1a, M2a, M2c, and the polymer center-of-mass distribution from full-monomer simulations (FMa); on the right we report the results for models M1b, M2b, and the polymer distribution function associated with the polymer midpoint (FMb).

Distribution *P*(σ, Φ) of for Φ = 1.09 and 4.36 for CG models M2a, M2b, and M2c and for polymers (FM). In the insets we report the deviations Δ*S* _{ g } = 100(*S* _{ g }/*S* _{ g, FM } − 1), where *S* _{ g } is the ratio (23) for the CG models and *S* _{ g, FM } is the corresponding quantity for polymers.

Distribution *P*(σ, Φ) of for Φ = 1.09 and 4.36 for CG models M2a, M2b, and M2c and for polymers (FM). In the insets we report the deviations Δ*S* _{ g } = 100(*S* _{ g }/*S* _{ g, FM } − 1), where *S* _{ g } is the ratio (23) for the CG models and *S* _{ g, FM } is the corresponding quantity for polymers.

## Tables

Virial-coefficient universal combinations for the models introduced in Sec. III and for the tetramer model (t) of Ref. 22. We also report the universal asymptotic values for polymers (p).^{27}

Virial-coefficient universal combinations for the models introduced in Sec. III and for the tetramer model (t) of Ref. 22. We also report the universal asymptotic values for polymers (p).^{27}

Compressibility factor *Z*(Φ) for the models introduced in Sec. III, for the tetramer model (t) of Ref. 22, and for polymers (p) in the scaling limit.^{41}

Compressibility factor *Z*(Φ) for the models introduced in Sec. III, for the tetramer model (t) of Ref. 22, and for polymers (p) in the scaling limit.^{41}

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