^{1,a)}and Andrea Pelissetto

^{2,b)}

### Abstract

We compare different coarse-grained single-blob models for star polymers. We find that phenomenological models inspired by the Daoud-Cotton theory reproduce quite poorly the thermodynamics of these systems, even if the potential is assumed to be density dependent, as done in the analysis of experimental results. Using the numerically determined coarse-grained potential, we also determine the minimum value *f* _{ c } of the functionality of the star polymer for which a fluid-solid transition occurs. By applying the Hansen-Verlet criterion we find 35 < *f* _{ c } ≲ 40. This result is confirmed by an analysis that uses the modified (reference) hypernetted chain method and is qualitatively consistent with previous work.

We thank Giuseppe D’Adamo for useful comments.

I. INTRODUCTION

II. THE EFFECTIVE PAIR POTENTIALS: DEFINITIONS

III. RESULTS: STRUCTURE AND THERMODYNAMICS

A. Zero-density results

B. Integral-equation methods

C. Comparing the potentials at finite density

D. Effective potentials with density-dependent corona diameter

IV. STARPOLYMERPHASE DIAGRAM

V. CONCLUSIONS

### Key Topics

- Polymers
- 24.0
- Binary stars
- 5.0
- Numerical modeling
- 5.0
- Phase diagrams
- 5.0
- Solution thermodynamics
- 5.0

## Figures

Comparison of the CG potentials of models MHG, M1, and M2: (top) *f* = 5; (bottom) *f* = 30. For *f* = 30 we fix τ = 1 in model M2 (see Sec. III C ).

Comparison of the CG potentials of models MHG, M1, and M2: (top) *f* = 5; (bottom) *f* = 30. For *f* = 30 we fix τ = 1 in model M2 (see Sec. III C ).

Relative deviations Δ*Z* _{ HNC } = *Z* _{ HNC }/*Z* _{ RY } − 1 and Δ*Z* _{ MHNC } = *Z* _{ MHNC }/*Z* _{ RY } − 1 for *f* = 30 and model MHG. The virial route (HNC,v and MHNC,v) and the compressibility route (HNC,c and MHNC,c) have been used.

Relative deviations Δ*Z* _{ HNC } = *Z* _{ HNC }/*Z* _{ RY } − 1 and Δ*Z* _{ MHNC } = *Z* _{ MHNC }/*Z* _{ RY } − 1 for *f* = 30 and model MHG. The virial route (HNC,v and MHNC,v) and the compressibility route (HNC,c and MHNC,c) have been used.

Relative deviation Δ*Z* = *Z*/*Z* _{ HG } − 1 for model M1 (* f* = 10, 18, 30) (top) and model M2 (* f* = 2, 5, 18, 30) (bottom) as a function of the polymer volume fraction Φ. *Z* _{ HG } is the compressibility factor obtained by using potential (2) .

Relative deviation Δ*Z* = *Z*/*Z* _{ HG } − 1 for model M1 (* f* = 10, 18, 30) (top) and model M2 (* f* = 2, 5, 18, 30) (bottom) as a function of the polymer volume fraction Φ. *Z* _{ HG } is the compressibility factor obtained by using potential (2) .

Structure factor as a function of *qR* _{ g } for Φ = 1: (a) *f* = 2 (models M2, MHG); (b) *f* = 5 (models M2, MHG); (c) *f* = 18 (models M1, M2, MHG); (d) *f* = 30 (models M1, M2, MHG); for *f* = 18, 30 the results for model M2 are obtained by setting τ = 1.

Structure factor as a function of *qR* _{ g } for Φ = 1: (a) *f* = 2 (models M2, MHG); (b) *f* = 5 (models M2, MHG); (c) *f* = 18 (models M1, M2, MHG); (d) *f* = 30 (models M1, M2, MHG); for *f* = 18, 30 the results for model M2 are obtained by setting τ = 1.

Log-log plot of the density-dependent corona diameter σ(Φ)/*R* _{ g } for *f* = 18 and *f* = 30. The solid line corresponds to a behavior Φ^{−3/4}.

Log-log plot of the density-dependent corona diameter σ(Φ)/*R* _{ g } for *f* = 18 and *f* = 30. The solid line corresponds to a behavior Φ^{−3/4}.

Compressibility factor *Z* as a function of Φ (left) and structure factors as a function of *qR* _{ g } for Φ = 1 (right). Data for model M1 with the density-dependent corona diameter reported in Fig. 5 and for model MHG. (Top) *f* = 18; (bottom) *f* = 30.

Compressibility factor *Z* as a function of Φ (left) and structure factors as a function of *qR* _{ g } for Φ = 1 (right). Data for model M1 with the density-dependent corona diameter reported in Fig. 5 and for model MHG. (Top) *f* = 18; (bottom) *f* = 30.

Structure factor maximum *S* _{max} = max *S*(*q*) as a function of Φ for *f* = 30, 35, 40. We use the MHG model and the RY closure.

Structure factor maximum *S* _{max} = max *S*(*q*) as a function of Φ for *f* = 30, 35, 40. We use the MHG model and the RY closure.

Effective hard-sphere packing fraction η_{ HS } determined by using the MHNC closure as a function of Φ. The horizontal line corresponds to the boundary of the fluid-solid coexistence η_{ HS } = 0.49.

Effective hard-sphere packing fraction η_{ HS } determined by using the MHNC closure as a function of Φ. The horizontal line corresponds to the boundary of the fluid-solid coexistence η_{ HS } = 0.49.

## Tables

In the second, third, and fourth columns we report virial coefficient ratios computed by using potential (HG). In the last three columns we report literature values obtained from full-monomer (FM) simulations. For a review of older estimates of *A* _{2}, see Ref. ^{ 33 } .

In the second, third, and fourth columns we report virial coefficient ratios computed by using potential (HG). In the last three columns we report literature values obtained from full-monomer (FM) simulations. For a review of older estimates of *A* _{2}, see Ref. ^{ 33 } .

Estimates of σ/*R* _{ g }, *A* _{2}, and *A* _{3} for models M1 and M2.

Estimates of σ/*R* _{ g }, *A* _{2}, and *A* _{3} for models M1 and M2.

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