^{1}, Jing Zong

^{1}, Delian Yang

^{1}and Qiang Wang

^{1,a)}

### Abstract

To highlight the importance of quantitative and parameter-fitting-free comparisons among different models/methods, we revisited the comparisons made by Groot and Madden [J. Chem. Phys.108, 8713 (Year: 1998)10.1063/1.476300] and Chen et al. [J. Chem. Phys.122, 104907 (Year: 2005)10.1063/1.1860351] between their dissipative particle dynamics (DPD) simulations of the DPD model and the self-consistent field (SCF) calculations of the “standard” model done by Matsen and Bates [Macromolecules29, 1091 (Year: 1996)10.1021/ma951138i] for diblock copolymer (DBC) A-B melts. The small values of the invariant degree of polymerization used in the DPD simulations do not justify the use of the fluctuation theory of Fredrickson and Helfand [J. Chem. Phys.87, 697 (Year: 1987)10.1063/1.453566] by Groot and Madden, and their fitting between the DPD interaction parameters and the Flory-Huggins χ parameter in the “standard” model also has no rigorous basis. Even with their use of the fluctuation theory and the parameter-fitting, we do not find the “quantitative match” for the order-disorder transition of symmetric DBC claimed by Groot and Madden. For lamellar and cylindrical structures, we find that the system fluctuations/correlations decrease the bulk period and greatly suppress the large depletion of the total segmental density at the A-B interfaces as well as its oscillations in A- and B-domains predicted by our SCF calculations of the DPD model. At all values of the A-block volume fractions in the copolymer f (which are integer multiples of 0.1), our SCF calculations give the same sequence of phase transitions with varying χN as the “standard” model, where N denotes the number of segments on each DBC chain. All phase boundaries, however, are shifted to higher χN due to the finite interaction range in the DPD model, except at f = 0.1 (and 0.9), where χN at the transition between the disordered phase and the spheres arranged on a body-centered cubic lattice is lower due to N = 10 in the DPD model. Finally, in 11 of the total 20 cases (f-χN combinations) studied in the DPD simulations, a morphology different from the SCF prediction was obtained due to the differences between these two methods.

We thank Professor Mark Matsen for providing us with his SCF results for the “standard” model, and Q.W. thanks Professor (Dr.) Kurt Kremer for the hospitality provided during his sabbatical stay at the Max Planck Institute for Polymer Research in Mainz, Germany. Financial support for this work was provided by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-07ER46448.

I. INTRODUCTION

II. MODEL AND METHODS

A. Model system used in DPD simulations of DBC

B. Fast off-lattice Monte Carlo (FOMC) simulations

C. Self-consistent field (SCF) calculations

III. RESULTS AND DISCUSSION

A. Symmetric DBC

1. Order-disorder transition(ODT)

2. Lamellar period

3. Segmental density profiles

B. Asymmetric DBC

1. SCF results

2. FOMC results

C. Comparing morphologies in SCF calculations and DPD simulations

IV. SUMMARY

### Key Topics

- Mean field theory
- 28.0
- Order disorder phase transitions
- 16.0
- Field theory models
- 9.0
- Lamellae
- 8.0
- Copolymers
- 6.0

##### C08F2/00

## Figures

Logarithmic plot of the bulk lamellar period L 0 of symmetric DBC as a function of χN. The SCF results are shown in curves, where “Standard” denotes the “standard” model, and the FOMC results for the DPD model are shown in symbols.

Logarithmic plot of the bulk lamellar period L 0 of symmetric DBC as a function of χN. The SCF results are shown in curves, where “Standard” denotes the “standard” model, and the FOMC results for the DPD model are shown in symbols.

Comparisons of the ensemble-averaged profiles of (a) the relative fraction of A segments in the direction (denoted by x) perpendicular to the lamellar interfaces, , and (b) the total segmental density in the x-direction, ϕ(x) ≡ ⟨ϕA(x) + ϕB(x)⟩, obtained from our SCF calculations and FOMC simulations of the DPD model at χN = 30π ( in the simulations), where the bulk lamellar period L 0/R e, 0 = 2.658 in the SCF calculations and 2.497 ± 0.002 in the FOMC simulations.

Comparisons of the ensemble-averaged profiles of (a) the relative fraction of A segments in the direction (denoted by x) perpendicular to the lamellar interfaces, , and (b) the total segmental density in the x-direction, ϕ(x) ≡ ⟨ϕA(x) + ϕB(x)⟩, obtained from our SCF calculations and FOMC simulations of the DPD model at χN = 30π ( in the simulations), where the bulk lamellar period L 0/R e, 0 = 2.658 in the SCF calculations and 2.497 ± 0.002 in the FOMC simulations.

Differences in the free energy per chain βf c of various ordered phases from that of the disordered phase obtained from our SCF calculations of the DPD model at (a) f = 0.4 and (b) f = 0.3. The inset of part (a) uses the free energy per chain of S, , as the reference, and the inset of part (b) shows the difference in the free energy per chain between L and C.

Differences in the free energy per chain βf c of various ordered phases from that of the disordered phase obtained from our SCF calculations of the DPD model at (a) f = 0.4 and (b) f = 0.3. The inset of part (a) uses the free energy per chain of S, , as the reference, and the inset of part (b) shows the difference in the free energy per chain between L and C.

(a) Reprinted Fig. 9 with permission from R. D. Groot and T. J. Madden, J. Chem. Phys.108, 8713 (Year: 1998)10.1063/1.476300. . (b) Hexagonal packing of cylinders in any x-y cross section of the cubic simulation box shown in part (a). (c) Reprinted lower-right panel of Fig. 6 with permission from R. D. Groot and T. J. Madden, J. Chem. Phys.108, 8713 (Year: 1998)10.1063/1.476300. . (d) Hexagonal packing of cylinders in any x-y cross section of the cubic simulation box shown in part (c). See text for more details.

(a) Reprinted Fig. 9 with permission from R. D. Groot and T. J. Madden, J. Chem. Phys.108, 8713 (Year: 1998)10.1063/1.476300. . (b) Hexagonal packing of cylinders in any x-y cross section of the cubic simulation box shown in part (a). (c) Reprinted lower-right panel of Fig. 6 with permission from R. D. Groot and T. J. Madden, J. Chem. Phys.108, 8713 (Year: 1998)10.1063/1.476300. . (d) Hexagonal packing of cylinders in any x-y cross section of the cubic simulation box shown in part (c). See text for more details.

## Tables

Mean-field ODT and the corresponding bulk lamellar period (shown in parentheses) of symmetric DBC for four combinations of chain model with non-bonded interaction potential, obtained from RPA. The upper-left corresponds to the “standard” model and the lower-right to the DPD model system.

Mean-field ODT and the corresponding bulk lamellar period (shown in parentheses) of symmetric DBC for four combinations of chain model with non-bonded interaction potential, obtained from RPA. The upper-left corresponds to the “standard” model and the lower-right to the DPD model system.

ODT χ*N and the corresponding bulk lamellar period of symmetric DBC for the DPD model at given segment number density ρ0σ3, obtained from FOMC simulations. The corresponding values of the invariant degree of polymerization , the Flory-Huggins parameter χ e , and that corrected for the fluctuation effects based on FH theory χ e, f are also listed. See main text for more details.

ODT χ*N and the corresponding bulk lamellar period of symmetric DBC for the DPD model at given segment number density ρ0σ3, obtained from FOMC simulations. The corresponding values of the invariant degree of polymerization , the Flory-Huggins parameter χ e , and that corrected for the fluctuation effects based on FH theory χ e, f are also listed. See main text for more details.

Comparisons of the bulk period from SCF calculations L 0, MF, that from variable-box FOMC simulations L 0, and the period from fixed-box simulations L for lamellae (at f = 0.5 and 0.4) and hexagonally packed cylinders (at f = 0.3) obtained with various parameters.

Comparisons of the bulk period from SCF calculations L 0, MF, that from variable-box FOMC simulations L 0, and the period from fixed-box simulations L for lamellae (at f = 0.5 and 0.4) and hexagonally packed cylinders (at f = 0.3) obtained with various parameters.

Stable regions in χN of various ordered phases obtained from the SCF calculations of the DPD and the “standard” models; results for the latter model are provided by Mark Matsen. For each A-block volume fraction in the copolymer f, the stable phase (having the lowest Helmholtz free energy per chain) is replaced by that in a lower row at higher χN. The last column lists the χN-values at which the DPD simulations 15,17 were performed; the morphology obtained from the DPD simulations is given in parentheses if it is different from the SCF prediction. See main text for more details.

Stable regions in χN of various ordered phases obtained from the SCF calculations of the DPD and the “standard” models; results for the latter model are provided by Mark Matsen. For each A-block volume fraction in the copolymer f, the stable phase (having the lowest Helmholtz free energy per chain) is replaced by that in a lower row at higher χN. The last column lists the χN-values at which the DPD simulations 15,17 were performed; the morphology obtained from the DPD simulations is given in parentheses if it is different from the SCF prediction. See main text for more details.

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