Microphase Separation and Phase Diagram of Concentrated Diblock Copolyelectrolyte Solutions Studied by Self-Consistent Field Theory Calculations in Two-Dimensional Space

Abstract

The self-consistent field theory (SCFT) is applied to the microphase separation of concentrated solutions of weakly charged polyelectrolytes. The generalized Poisson–Boltzmann equation describing the electrostatic interactions at the mean-field level is numerically solved by a full multigrid algorithm, which enables one to solve the SCFT equations of polyelectrolyte systems in real space as efficient as neutral polymer systems. To demonstrate the power of the real-space numerical scheme, we consider a diblock copolyelectrolyte consisting of a charged block and a neutral block in two-dimensional space. The phase diagram in the Flory–Huggins interaction parameterthe composition space is constructed by numerical calculations. The density distribution of polymer segments, the counterions, and the net charge of the ordered structures, namely the lamellar phase and the hexagonal phase, are intensively examined. The effects of the interaction parameter and the degree of ionization are examined carefully. The numerical scheme can be easily extended to 3D calculations, various chain architectures, various charge distribution models, and other statistics chain models such as the worm-like chain model without losing any computational efficiency.

Self-consistent field theory study of the solvation effect in polyelectrolyte solutions: beyond the Poisson–Boltzmann model

“…For example, Yang et al recently proposed a highly efficient multi-grid method, but there is no standard solver for this method. 23 The present method is also highly efficient as long as the f(r) is smooth enough. More important, it can be easily solved by employing the FFTW packages.…”

“…Therefore, the system studied becomes a salt-free solution of charged-neutral diblock copolymers, which is another system that researchers are interested in. 23 Fig. 6 shows the dependence of the order parameter proles on p for the charged-neutral diblock copolymer solutions with a B ¼ 0.08.…”

“…The SCFT approach for polyelectrolyte systems was rst proposed by Borukhov et al for semidilute solutions of polyelectrolytes and polyampholytes possessing various charge distributions, 17,18 and further generalized by Shi and Noolandi to multicomponent polyelectrolyte systems. 19 This approach was also applied to study the phase separation of block polyelectrolytes, [20][21][22][23] adsorption of polyelectrolytes on a charged surface, [24][25][26] polymer brushes, 27,28 connement effects, 29 counterion adsorption, 30 and so on.…”

Conformation of a single polyelectrolyte in poor solvents

“…The smeared charge model is adopted to describe the charge distribution on the chain backbone such that the annealed fraction of segments α contains a charge with valency zp. [62][63][64][65][66][67][68] While the choice of charge is arbitrary, we choose the charge on the PE backbone to be negative here. Mobile ions are taken as point charges with valency z±.…”

Theory of Microphase Separation in Concentrated Solutions of Sequence-Specific Charged Heteropolymers

“…Following our earlier work, − we integrate the degrees of freedom of counterions and salt ions and obtain the contribution from the translational entropy of dissociated ions and also derive the Debye–Hückel (DH) interaction, − , which is free from any artificial mathematical divergences. Furthermore, since we treat only the weak segregation limit in the present study, we have taken the distribution of small ions to be uniform as suggested by the results from RPA. , …”