Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results

Abstract

For conformationally asymmetric diblock copolymer (DBC) melts, we proposed a model system that can be readily used in dissipative particle dynamics (DPD) simulations and performed the corresponding self-consistent field (SCF) calculations to study the stability of Frank−Kasper (FK) phases. This provides the mean-field results needed for direct comparison with the DPD simulations, without any parameter fitting, to unambiguously quantify the fluctuation/correlation effects inherently neglected by the SCF theory but important in the low-molecular-weight DBC melts forming FK phases in experiments. Our SCF calculations of the DPD model are faster and use less memory, both by at least one order of magnitude, than those of the "standard" model and give accurate results. Among the five FK (i.e., A15, C14, C15, Z, and σ) phases considered here, C14 is found to be the only stable one; its stability is due to the conformational asymmetry and, in particular, it is stabilized over other FK phases by entropy. This is the first time C14 is predicted to be stable in SCF calculations of neat DBC melts. Our SCF phase diagrams for the DPD model of conformationally asymmetric DBCs are qualitatively different from those for the "standard" model reported in the literature, which may be due to the model differences and numerical accuracy.

Correction to “Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results”

“…For ε > 1, however, our phase diagrams are qualitatively different from those reported in ref ; in particular, σ instead of C14 becomes the only stable FK phase, and the phase boundary between S b and S f and that between S f and the disordered phase (D) at f = 0.2 are also altered. On the other hand, our previous results that all phase boundaries are shifted to smaller χ N as f increases from 0.2 to 0.3 and that a small region of S f exists at f = 0.2 but disappears at f = 0.3 are still valid. Finally, we note that within our parameter range, A15 is not stable for the DPD model at N = 10.…”

“…We see that the stability of σ at intermediate χ N is due to its delicate balance between β u c ,χ (or β u c = β u c ,χ + β u c ,κ ) and s c / k B = s c ,A / k B + s c ,B / k B , and that of C at larger χ N is mainly due to its smallest β u c ,χ (or β u c ) and − s c ,B / k B ; in contrast, S b at smaller χ N is mainly stabilized by its smallest − s c ,A / k B (or − s c / k B ). In addition, our results show that β u c ,κ accounts for only 0.3∼0.5% of β u c , thus indicating that our DPD model at N /κ = 50π is nearly incompressible as demonstrated in ref . Figure replaces that in ref ; note that the χ N -ranges in these two figures are different due to the corrected phase diagram.…”

“…In addition, our results show that β u c ,κ accounts for only 0.3∼0.5% of β u c , thus indicating that our DPD model at N /κ = 50π is nearly incompressible as demonstrated in ref . Figure replaces that in ref ; note that the χ N -ranges in these two figures are different due to the corrected phase diagram.…”

“…In a recent paper, we performed the self-consistent field (SCF) calculations of a dissipative particle dynamics (DPD) model of conformationally asymmetric diblock copolymer A-B melts to study the stability of various Frank–Kasper (FK) phases formed by such A-B melts. While all the equations in ref are correct, we did not calculate the dimensionless (mean-field) Helmholtz free energies per chain β f c = β u c ,χ + β u c ,κ – s c / k B and the dimensionless stresses dβ f c /d θ of ordered phases accurately.…”

“…In a recent paper, we performed the self-consistent field (SCF) calculations of a dissipative particle dynamics (DPD) model of conformationally asymmetric diblock copolymer A-B melts to study the stability of various Frank–Kasper (FK) phases formed by such A-B melts. While all the equations in ref are correct, we did not calculate the dimensionless (mean-field) Helmholtz free energies per chain β f c = β u c ,χ + β u c ,κ – s c / k B and the dimensionless stresses dβ f c /d θ of ordered phases accurately. For example, the dimensionless (nonbonded) internal energy per chain due to the system compressibility β u c,κ = ( N /2κ)∑ q β û 0 (| q |)(ϕ̂ A ( q ) + ϕ̂ B ( q ) – δ q , 0 )(ϕ̂ A (− q ) + ϕ̂ B (− q ) – δ – q , 0 ) should be calculated in the reciprocal space as given (note that β û 0 (| q |) is obtained analytically, and that since ϕ A ( r ) is real ϕ̂ A false( prefix− boldq false) = ϕ̂ A ̅ false( boldq false) , i.e., the complex conjugate of ϕ̂ A ( q )); this corresponds to numerically evaluating β u c,κ = ( N /2κ)∫d r (ϕ A ( r ) + ϕ B ( r ) – 1)ψ( r ) with ψ( r ) ≡ (∫d r ′β u 0 (| r – r ′|)(ϕ A ( r ′) + ϕ B ( r ′) – 1)/ V in the real space via the (composite) trapezoidal rule.…”